Non-GRS type Euclidean and Hermitian LCD codes and Their Applications for EAQECCs

Non-GRS t yp e Euclidean and Hermitian LCD co des and Their App lications for EA QECCs ∗ Zhonghao Liang 1 , Dongmei Huang 1 , Qunyi ng Liao † 1 , Cuiling F an 2 , and Zhengc h un Zhou 3 1 College of Mathematic al Sciences, Sic h u an Normal Univ ersit y , Chengdu 610066, Chin a 2 Sc ho ol of Mathematics, Southw est Jiaotong Universit y , Chengdu 6117 56, Ch ina 3 Sc ho ol of Information Science and T ec h nology , Southw est J iaotong Univ ersit y , Chengdu, 6117 56, China Abstract. In recent y ears, the construction of non-GRS t yp e linear cod es h as attracted con- siderable atten tion d ue to that they can effectiv ely resist b oth the S idelnik o v-Sh estak o v attac k and the Wiesc hebr ink attac k. Constructing linear complemen tary dual (LCD) cod es and determining the h ull of linear co des hav e long b een imp ortan t topics in co d in g theory , as they pla y the cru cial role in constructing en tangle men t-assisted qu antum error-correcting cod es (EAQECCs), certai n comm u- nication systems and cryp tograph y . In this pap er, by utilizing a class of non-GRS t yp e linear co des, namely , generalized Roth-Lemp el (in short, GRL) co des, we firstly construct several classes of Eu- clidean LCD codes, Hermitian LCD codes, and linear codes with small-dimensional hulls, ge neralized the main results giv en b y W u et al. in 2021. W e also p resen t an u pp er b oun d for the num b er of a class of Euclidean GRL cod es with 1-dimensional h ull, and then for sev eral classes of Hermitian GRL co des, we firstly deriv e an upp er b ound for the dimension of the hull, and pr o ve that the b ou n d is attainable. Secondly , as an application, w e obtain sev eral families of EA QECCs. Thirdly , w e pr o ve that the GRL co de is non-GRS for k > ℓ . Finally , some corresp onding examples for LCD MDS cod es and LCD NMDS co des are presented. Keyw ords. Roth-Lemp el co des; LCD co des; Hulls; En tanglemen t-assisted q u an tum error- correcting co des. 1 In tro duc t ion Let F q b e a finite field with q elem en ts. An [ n, k , d ] linear code C ov er F q is a linear subspace of F n q with dimension k and minim um Hamming distance d . The Euclidean dual co de and the Hermitian dual co de of C are resp ectiv ely defined by C ⊥ E = ( ( x 1 , . . . , x n ) = x ∈ F n q | h x , y i E = n X i =1 x i y i = 0 , ∀ y = ( y 1 , . . . , y n ) ∈ C ) ∗ E-mails: lia ngzhongh08 07@16 3.com; 31201 93984 @qq.com; qunyingliao@sicnu.edu.cn; cuilingfan@163.co m ; zzc@swjtu.edu.cn † Corresp onding author 1 and C ⊥ H = ( ( x 1 , . . . , x n ) = x ∈ F n q 2 | h x , y i H = n X i =1 x i y q i = 0 , ∀ y = ( y 1 , . . . , y n ) ∈ C ) . F or a linear co de C , the hull is defined by Hull ( C ) = C ∩ C ⊥ , where C ⊥ is the dual co de of C . The h ull pla ys an imp ort a n t role for determining the complexit y of algorithms to c hec k the p erm utation equiv alence of t w o linear co des[ 34 ], computing the automor phism group of a linear co de[ 20 ], calculating the num b er of shared pairs that r equired t o construct an en ta nglemen t-a ssisted quan tum error-correcting co de (EA QECC)[ 10 ], and so on. In particular, these alg orithms tend to b e hig hly effectiv e when the dimension o f the h ull is small. In addition, it is w o rth men tioning tha t a sp ecial case o f the h ull of linear co des is of m uc h in terest, i.e., Hull ( C ) = { 0 } , in whic h C is called a linear complemen tary dual (in short, LCD) co de. And LCD co des ha v e b een widely a pplied in data storage, commu nication systems, and cryptograph y[ 1 , 5 , 33 ]. Thus , determining the v alue of dim ( Hull ( C ) ), constructing LCD co des or linear co des with low-dimens ional hulls has b een in teresting [ 6 , 7 , 1 5 , 1 6 , 18 , 2 1 , 2 5 , 37 ]. The w ell-known Singleton Bound sa ys that d ≤ n − k + 1 for any [ n, k , d ] co de C o v er F q , whic h means that S ( C ) = n + 1 − k − d is an non-negativ e in teger. If S ( C ) = 0, then the co de C is maximum distance separable (in short, MDS). If S ( C ) = 1, then the co de C is almost MDS (in short, AMDS). Esp ecially , if S ( C ) = S  C ⊥  = 1, then C is near MDS (in short, NMDS). The most w ell-kno wn class of MDS co des is the generalized Reed-Solomon (in short, GRS) co de. If the co de C is not equiv alent to an y GRS co de, then the co de C is called to b e non-GRS t ype. Note that MDS co des constructed from GR S co des are equiv alent to GRS co des, and so an MDS co de is either a GRS type or a non-GRS t yp e, a s the following F igure 1 . GRS type non-GRS type MDS c o des Figure 1: Classification of MDS co des Up to no w, a lot of results ha ve b een obtained o n constructing non-GRS t yp e linear co des[ 2 – 4 , 1 3 , 23 , 24 , 31 , 35 , 38 ]. So far, most of LCD co des and EA QECCs hav e b een constructed from a GRS type linear co de [ 6 , 9 , 14 , 15 , 17 , 18 , 22 , 25 , 42 ]. Th us, constructing LCD co des or EA QECCs is in teresting based o n non-GRS ty p e linear co des. There hav e already b een some results o n constructing Euclidean LCD co des based on non-GRS t yp e linear co des[ 11 , 19 , 29 , 36 , 40 ]. While, for constructing Hermitian LCD co des and EA QECCs based on non-GRS t ype linear co des, there are few relev ant work [ 30 , 36 ]. Esp ecially , in 202 1, W u et al. [ 36 ] considered the Roth-Lemp el (in short, RL) co de prop osed b y Roth R M. and Lemp el A.[ 31 ] in 19 89, whic h has the f o llo wing generator matrix  G RS ( α ) 0 ( k − 2) × 2 T 2 × 2 ( δ )  k × ( n +2) , 2 where G RS ( α ) is the generator matrix of the RS co de with the ev aluation-p oin t sequence α = ( α 1 , . . . , α n ) ∈ F n q , and T 2 × 2 ( δ ) =  0 1 1 δ  with δ ∈ F q . By ta king some sp ecial α , they pro v ed that there exists Hermitian LCD RL codes. In 2025, the a uthors[ 26 ] in tro duced the generalized Roth-Lemp el (in short, GRL) co de, whic h is a generalization of RL co des, the corresp onding linear co de o ver F q has the g enerato r mat r ix  G RS ( α ) 0 ( k − ℓ ) × ℓ A ℓ × ℓ  k × ( n + ℓ ) , where G RS ( α ) is the generator matrix of the RS co de with the ev aluation-p oin t sequence α = ( α 1 , . . . , α n ) ∈ F n q , and A ℓ × ℓ ∈ GL l ( F q ). So far, fo r sev eral sp ecial matrices A ℓ × ℓ , the corresp onding MDS prop erty [ 26 , 38 ], AMDS prop erty[ 26 , 38 ], NMDS prop erty[ 12 , 2 7 , 39 ], self-dual prop erty[ 26 , 38 ], extendable prop erty [ 28 , 38 ], the existence of LCD co des[ 36 ] a nd deco ding algorithms of punctured co des [ 43 ] ha ve b een inv estigated, resp ectiv ely . In this pa p er, differen t from the w o r k of W u et al. ( 2021), whic h only considered a v ery special class of 2 × 2 matrices T 2 × 2 ( δ ) and some sp ecial α , a nd they o nly pro ved that there exists LCD RL co des, w e extend their researc h to the most general ℓ × ℓ matrix A ℓ × ℓ and more flexible α , a nd give the specific construction of LCD codes. W e construct sev eral classes of Euclidean LCD co des, Hermitian LCD co des, small-dimensional h ull linear co des and EA QECCs. And for some Hermitian GRL co des, w e also obtain an upp er b ound for the dimension of t he hull a nd pro v e that the b ound is attainable. This pap er is organized as follows. In Section 2 , w e give the definition of the G R L co de and some necessary lemmas. In Sections 3 - 4 , we giv e some constructions for Euclidean LCD GRL co des, Hermitian LCD GR L co des, small-dimensional hull linear co des and EA QECCs . In Section 5 , we prov e that the G R L co de is non- G RS. In Section 6 , w e conclude the whole pap er. In App endix, some examples are given. 2 Preliminaries Throughout this pap er, for the conv enience, w e fix some nota t io ns a s the fo llowing. • q is a p o w er of an o dd prime. • F q or F q 2 is the finite field with q or q 2 elemen ts, resp ectiv ely . • F or any non-empty set { x 1 , . . . , x n } , { x 1 , . . . , x n } mod k , { x 1 ( mo d k ) , . . . , x n ( mo d k ) } . • F or an y set A , # A denotes t he num b er o f elemen ts in A . • d ( x ) denotes the num b er of p ositiv e divisors of the p o sitiv e integer x . • F or the prime num b er p and the p o sitiv e in teger x , v p ( x ) denotes the largest non-negativ e in teger k suc h that p k | x and p k +1 ∤ x . • gcd ( a, b ) denotes the great est common divisor for t w o p ositiv e integers a and b . • dim (Hull E ( C )) o r dim (Hull H ( C )) denotes the dimension o f the Euclidean hu ll or the Hermitian h ull for the linear co de C , resp ectiv ely . 3 • F or an y matrix G , G denotes the conjug ate matrix of G . In this section, w e recall the definition of the generalized Roth-Lemp el co de and some necessary lemmas. Definition 2.1 ( [ 26 ], Definition 1) L et 3 ≤ ℓ + 1 ≤ k + 1 ≤ n ≤ q , α = ( α 1 , . . . , α n ) ∈ F n q with α i 6 = α j ( i 6 = j ) and v = ( v 1 , . . . , v n ) ∈  F ∗ q  n . The gener alize d R oth-L emp el (in short, GRL) c o de GRL k ( α , v , A ℓ × ℓ ) is define d as GRL k ( α , v , A ℓ × ℓ ) ,  ( v 1 f ( α 1 ) , . . . , v n f ( α n ) , β ) | f ( x ) ∈ F k q [ x ]  , wher e A ℓ × ℓ = ( a ij ) ℓ × ℓ ∈ G L ℓ ( F q ) an d β = ( f k − ℓ , . . . , f k − 1 ) A ℓ × ℓ =  a 11 f k − ℓ + a 21 f k − ( ℓ − 1) + · · · + a ℓ 1 f k − 1 , . . . , a 1 ℓ f k − ℓ + a 2 l f k − ( ℓ − 1) + · · · + a ℓℓ f k − 1  . Lemma 2.1 ( [ 38 ], D efinition 3) L et C 1 and C 2 b e two l i n e ar c o des of the sam e c o de leng th over F q , and let M b e a gener ator ma trix of C 1 . Then C 1 and C 2 ar e monomial ly e quivalent if and only if ther e ex i s ts a monom i al matrix D such that M D is a gener ator matrix of C 2 . Remark 2.1 By Definition 2.1 , the c o de GRL k ( α , v , A ℓ × ℓ ) and the c o de GRL k ( α , 1 , A l × l ) have the gener ator matrix G v ,n =            v 1 · · · v n 0 · · · 0 v 1 α 1 · · · v n α n 0 · · · 0 . . . . . . . . . . . . . . . . . . v 1 α k − ( ℓ +1) 1 · · · v n α k − ( ℓ +1) n 0 · · · 0 v 1 α k − ℓ 1 · · · v n α k − ℓ n a 11 · · · a 1 ℓ . . . . . . . . . . . . . . . . . . v 1 α k − 1 1 · · · v n α k − 1 n a ℓ 1 · · · a ℓℓ            (2.1) and G 1 ,n =            1 · · · 1 0 · · · 0 α 1 · · · α n 0 · · · 0 . . . . . . . . . . . . . . . . . . α k − ( ℓ +1) 1 · · · α k − ( ℓ +1) n 0 · · · 0 α k − ℓ 1 · · · α k − ℓ n a 11 · · · a 1 ℓ . . . . . . . . . . . . . . . . . . α k − 1 1 · · · α k − 1 n a ℓ 1 · · · a ℓℓ            , (2.2) r esp e ctively. And so by L emma 2.1 , the c o d e G RL k ( α , v , A ℓ × ℓ ) and the c o de G RL k ( α , 1 , A l × l ) ar e monomial ly e quivalent. The follo wing Lemma 2 . 2 giv es a c haracterization for a linear co de to be an Euclidean LCD co de o r a Hermitian LCD co de. Lemma 2.2 ( [ 6 ], Prop osition 2) I f G is a gener a tor matrix for the [ n, k ] line ar c o de C over F q (r esp. F q 2 ), then C is an Euclide an (r esp. Hermitian) LCD c o de if a n d only if the k × k matrix GG T (r esp. G ) is nons i n gular over F q (r esp. F q 2 ). 4 The follo wing Lemma 2 . 3 is v ery imp ortant for our constructions. Lemma 2.3 ( [ 6 ]) Let F ∗ q = h γ i (resp. F ∗ q 2 = h γ i ), s b e a p ositive integer with s | q − 1(resp. s | q 2 − 1), a nd α i = γ q − 1 s i (resp. α i = γ q 2 − 1 s i ) for 1 ≤ i ≤ k , then for an y integer t and β ∈ F ∗ q (resp. β ∈ F ∗ q 2 ), we hav e s X i =1 ( β α i ) t =  β t s, if s | t ; 0 , otherwise . The follo wing Lemma 2 . 4 pro vides a metho d to determine whether a linear co de is non-RS t ype. Lemma 2.4 ( [ 32 ], Theorem 1) L et α = ( α 1 , . . . α n ) ∈ F n q with α i 6 = α j ( i 6 = j ) . Supp ose that B is a k × ( n − k ) matrix and G = ( E k | B ) is a k × n matrix over F q , wher e E k is the k × k identity m atrix. Then G gener ates the RS c o de R S k ( α ) if and only if for 1 ≤ i ≤ k and 1 ≤ j ≤ n − k , the ( i, j ) -th e ntry of B is given by η k + j η − 1 i α k + j − α i , wher e η i = k Q s =1 ,s 6 = i ( α i − α s ) an d η k + j = k Q s =1 ( α k + j − α s ) . The following Lemma 2.5 provides an explicit formula for the n um b er of k - tuples ( x 1 , . . . , x k ) ∈ F k q suc h t ha t x 2 1 + · · · + x 2 k = c ∈ F q holds. Lemma 2.5 ( [ 8 ]) F or any elemen t c ∈ F q , let v ( c ) =  q − 1 , c = 0; − 1 , c ∈ F ∗ q , and N f ( k , c, q ) denote the n umber of k - tuples ( x 1 , . . . , x k ) ∈ F k q \ { 0 } suc h that x 2 1 + · · · + x 2 k = c holds, then N f ( k , c, q ) =    q n − 1 + v ( b ) q k 2 − 1 η  ( − 1) k 2  , 2 | k ; q n − 1 + v ( b ) q k − 1 2 η  ( − 1) k − 1 2 c  , 2 ∤ k . In particular, for ( x 1 , . . . , x k ) ∈  F ∗ q  k , F eng et al. [ 41 ] ga v e the follo wing Lemma 2.6 . Lemma 2.6 ( [ 41 ], Theorems 2.6- 2.7) F or an y elemen t c ∈ F q , let N ∗ f ( k , c, q ) denote the n um b er of k -tuples ( x 1 , . . . , x k ) ∈  F ∗ q  k suc h that x 2 1 + · · · + x 2 k = c holds, then the follo wing statemen ts are true, (1) if q ≡ 1 (mo d 4), then N ∗ f ( k , c, q ) =                2( q − 1) k + ( q − 1)  ( √ q − 1 ) k + ( − 1 − √ q ) k  2 q , if c = 0; 2( q − 1) k + ( √ q − 1) k +1 + ( − 1 − √ q ) k +1 2 q , if c = a 2 for some a ∈ F ∗ q ; 2( q − 1) k + (1 − q )  ( √ q − 1 ) k − 1 + ( − 1 − √ q ) k − 1  2 q , otherwise; 5 (2) if q ≡ 3 (mo d 4), then N ∗ f ( k , c, q ) =                2( q − 1) k + ( q − 1)  ( √ − q − 1) k + ( − 1 − √ − q ) k  2 q , if c = 0; 2( q − 1) k + ( q + 1)  ( √ − q − 1) k − 1 + ( − 1 − √ − q ) k − 1  2 q , if c = a 2 for some a ∈ F ∗ q ; 2( q − 1) k + ( √ − q − 1) k +1 + ( − 1 − √ − q ) k +1 2 q , otherwise . F or an [ n, k , d ] linear co de, the fo llo wing Lemmas 2.7 - 2.10 pro vide a metho d for calculating the dimension of the Euclidean h ull or the Hermitian hull, and constructing EA QECCs. Lemma 2.7 ( [ 10 ], Prop osition 3.1) Let C b e an [ n, k , d ] q linear code with the generator matrix G and the parity che c k matrix H . Then, w e hav e rank  H H ⊥ E  = n − k − dim (Hull E ( C )) and rank  GG ⊥ E  = k − dim  Hull E  C ⊥ E  . Lemma 2.8 ( [ 10 ], Corollary 3.1) L et C a nd C ⊥ E b e the classical linear co de and its Euclidean dual code with the parameters [ n, k , d ] q and  n, k , d ⊥ E  q , resp ectiv ely . Then there exist t wo EA QECCs with the parameters [[ n, k − dim (Hull E ( C )) , d, n − k − dim (Hull E ( C ))]] q and  n, n − k − dim (Hull E ( C )) , d ⊥ E , k − dim ( Hull E ( C ))  q , resp ectiv ely . Moreov er, if C is MDS, then the ab o v e tw o EAQECC s are also MDS. Lemma 2.9 ( [ 10 ], Prop o sition 3.2) Let C b e the [ n, k , d ] q 2 linear co de with the generator matrix G and t he parity c hec k matrix H . Then, we hav e rank  H H ⊥ H  = n − k − dim (Hull H ( C )) and rank  GG ⊥ H  = k − dim  Hull H  C ⊥ H  . Lemma 2.10 ( [ 10 ], Corollary 3.2) Let C a nd C ⊥ H b e a classical linear co de and its Hermitian dual with the parameters [ n, k , d ] q 2 and  n, k , d ⊥ H  q 2 , r espective ly . Then there exists tw o EA QECCs with the parameters [[ n, k − dim (Hull H ( C )) , d, n − k − dim (Hull H ( C ))]] q and  n, n − k − dim (Hull H ( C )) , d ⊥ H , k − dim ( Hull H ( C ))  q , resp ectiv ely . Moreov er, if C is MDS, then the ab o v e tw o EAQECC s are also MDS. 6 3 Euclidean LCD GRL co des and the ir applications Throughout this section, w e fix k | q − 1, F ∗ q = F q \ { 0 } = h γ i and α i = γ q − 1 k i (1 ≤ i ≤ k ). In this section, b y taking a special v ector α = ( α 1 , . . . , α n ), w e construct four classes of Euclidean LCD GRL co des with the parameters [ n + ℓ, k ], get some GRL co des with small-dimensional h ull, and then obtain sev era l families of EA QECCs. 3.1 The first class of LCD GRL co des with the parameters [ k + ℓ, k ] In this subsection, b y ta king α =  γ δ α 1 , . . . , γ δ α k  , w e construct tw o classes of Euclidean LCD GRL co des, get a class of G RL co des with 1-dimensional h ull, and then obtain an upp er b ound for the num b er o f GRL co des with 1-dimensional h ull. Theorem 3.1 L et α =  γ δ α 1 , . . . , γ δ α k  with 1 ≤ δ ≤ q − 1 . Then the fol lowing two state- ments ar e true, (1) if ℓ < k 2 , then the c o de GRL k ( α , v , A ℓ × ℓ ) is Euclide an LCD; (2) if ℓ = k 2 and γ δk k + ℓ P i =1 a 2 1 i ∈ F ∗ q , then the c o de GRL k ( α , v , A ℓ × ℓ ) is Euclide an LCD. Pro of . By Remark 2.1 , w e only fo cus on G RL k ( α , 1 , A l × l ) with the generator matrix G 1 ,k giv en by ( 2 . 2 ). F urthermore, b y Lemma 2.2 , w e only need t o prov e that t he k × k matrix G 1 ,k G T 1 ,k is nonsingular o ver F q , i.e., rank  G 1 ,k G T 1 ,k  = k . In fact, note t ha t G 1 ,k G T 1 ,k =                          k k P i =1  γ δ α i  · · · k P i =1  γ δ α i  k − ℓ − 1 k P i =1  γ δ α i  k − ℓ · · · k P i =1  γ δ α i  k − 1 k P i =1  γ δ α i  k P i =1  γ δ α i  2 · · · k P i =1  γ δ α i  k − ℓ k P i =1  γ δ α i  k − ℓ +1 · · · k P i =1  γ δ α i  k . . . . . . . . . . . . . . . . . . . . . k P i =1  γ δ α i  k − ( ℓ +1) k P i =1  γ δ α i  k − ℓ · · · k P i =1  γ δ α i  2 k − 2 ℓ − 2 k P i =1  γ δ α i  2 k − 2 ℓ − 1 · · · k P i =1  γ δ α i  2 k − ℓ − 2 k P i =1  γ δ α i  k − ℓ k P i =1  γ δ α i  k − ℓ +1 · · · k P i =1  γ δ α i  2 k − 2 ℓ − 1 k P i =1  γ δ α i  2 k − 2 ℓ + ℓ P i =1 a 2 1 i · · · k P i =1  γ δ α i  2 k − ℓ − 1 + ℓ P i =1 a 1 i a ℓi . . . . . . . . . . . . . . . . . . . . . k P i =1  γ δ α i  k − 1 k P i =1  γ δ α i  k · · · k P i =1  γ δ α i  2 k − ℓ − 2 k P i =1  γ δ α i  2 k − ℓ − 1 + ℓ P i =1 a ℓi a 1 i · · · k P i =1  γ δ α i  2 k − 2 + ℓ P i =1 a 2 ℓi                          . 7 F or (1). By ℓ < k 2 , w e hav e 2 k − 2 ℓ > k , furthermore, by Lemma 2.3 , w e can get G 1 ,k G T 1 ,k =                         k 0 · · · 0 0 · · · 0 0 · · · 0 0 · · · 0 0 0 · · · 0 0 · · · 0 0 · · · 0 0 · · · γ δk k . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · 0 0 · · · 0 0 · · · 0 γ δk k · · · 0 0 0 · · · 0 0 · · · 0 0 · · · γ δk k 0 · · · 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · 0 0 · · · 0 γ δk k · · · 0 0 · · · 0 0 0 · · · 0 0 · · · γ δk k 0 · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · 0 γ δk k · · · 0 0 · · · 0 0 · · · 0 0 0 · · · γ δk k 0 · · · 0 0 · · · 0 ℓ P i =1 a 2 1 i · · · ℓ P i =1 a 1 i a ℓi . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 γ δk k · · · 0 0 · · · 0 0 · · · 0 ℓ P i =1 a ℓi a 1 i · · · ℓ P i =1 a 2 ℓi                         . No w from k | q − 1 and F ∗ q = h γ i , w e ha ve γ δk k ∈ F ∗ q , and so rank  G 1 ,k G T 1 ,k  = k . F urthermore, b y Lemma 2.2 , the co de G R L k ( α , v , A l × l ) is Euclide an LCD. F or (2). By ℓ = k 2 , w e can get 2 k − 2 ℓ = k , furthermore, b y Lemma 2.3 , w e hav e G 1 ,k G T 1 ,k =                    k 0 · · · 0 0 0 · · · 0 0 0 · · · 0 0 0 · · · γ δk k . . . . . . . . . . . . . . . . . . 0 0 · · · 0 0 γ δk k · · · 0 0 0 · · · 0 γ δk k + ℓ P i =1 a 2 1 i ℓ P i =1 a 1 i a 2 i · · · ℓ P i =1 a 1 i a ℓi 0 0 · · · γ δk k ℓ P i =1 a 2 i a 1 i ℓ P i =1 a 2 2 i · · · ℓ P i =1 a 2 i a ℓi . . . . . . . . . . . . . . . . . . 0 γ δk k · · · 0 ℓ P i =1 a ℓi a 1 i ℓ P i =1 a ℓi a 2 i · · · ℓ P i =1 a 2 ℓi                    . No w b y k | q − 1, F ∗ q = h γ i and γ δk k + ℓ P i =1 a 2 1 i ∈ F ∗ q , we can get r ank  G 1 ,k G T 1 ,k  = k , th us by Lemma 2.2 , the co de G R L k ( α , v , A ℓ × ℓ ) is Euclidean LCD . This completes the pro of o f Theorem 3 . 1 .  Remark 3.1 By taking δ = q − 1 and A l × l =  0 1 1 τ  with τ ∈ F q in The or em 3.1 (1) , the c orr esp ondi ng r esult is just L emma 3.5 (1) in [ 36 ]. By Lemma 2.7 , it’s easy to obtain the follo wing 8 Theorem 3.2 L et α =  γ δ α 1 , . . . , γ δ α k  , ℓ = k 2 and γ δk k + ℓ P i =1 a 2 1 i = 0 . Then dim (Hull E (GRL k ( α , v , A ℓ × ℓ ))) = 1 . The following Theorem 3 .3 presen ts an upp er b ound f o r the num b er o f the co de GRL k ( α , v , A ℓ × ℓ ) with 1-dimensional h ull in Theorem 3.2 . Theorem 3.3 I f α =  γ δ α 1 , . . . , γ δ α k  , ℓ = k 2 and k + ℓ P i =1 a 2 1 i = 0 . Then the fol lowing two statements a r e true, (1) if ( a 11 , . . . , a 1 ℓ ) ∈ F ℓ q \ { 0 } , then the numb er of the c o de GRL k ( α , v , A ℓ × ℓ ) with 1 - dimensional hul l is le s s than or e qual to  d  q − 1 2  − 1  · N f  ℓ, γ δk + q − 1 2 k , q  · ℓ − 1 Y i =1  q ℓ − q i  ; (2) if ( a 11 , . . . , a 1 ℓ ) ∈  F ∗ q  ℓ , then the numb er of the c o d e G R L k ( α , v , A ℓ × ℓ ) with 1 - dimensional hul l is le s s than or e qual to  d  q − 1 2  − 1  · N ∗ f  ℓ, γ δk + q − 1 2 k , q  · ℓ − 1 Y i =1  q ℓ − q i  . Pro of. By the equiv alence of linear co des, it’s easy to kno w that the n umber of the co de GRL k ( α , v , A ℓ × ℓ ) is less than or equal to the n umber of the matrix G 1 ,k giv en by ( 2 . 2 ). F urthermore, b y Theorem 3.2 , w e know that the n um b er of the co de GRL k ( α , v , A ℓ × ℓ ) with 1-dimensional h ull is less than or equal t o the n um b er of the matrix G 1 ,k satisfying the following three conditions sim ultaneously , (i) α =  γ δ α 1 , . . . , γ δ α k  with α i = γ q − 1 k i (1 ≤ i ≤ k ); (ii) k = 2 ℓ ; (iii) A l × l = ( a ij ) ℓ × ℓ ∈ G L ℓ ( F q ) with γ δk k + ℓ P i =1 a 2 1 i = 0. F or the conditions (i) a nd (ii), it is easy to kno w that fo r g iven q , γ and δ , the n um b er of the v ector α dep ends on the n um b er M of k , i.e., M =# n α =  γ δ α 1 , . . . , γ δ α k  : α i = γ q − 1 k i , 1 ≤ i ≤ k , k = 2 ℓ | q − 1 o =# { k : 2 ≤ ℓ ≤ k | q − 1 } =#  ℓ : 2 ≤ ℓ | q − 1 2  = d  q − 1 2  − 1 . F or the condition (iii), for the conv enience, let a i (1 ≤ i ≤ ℓ ) b e the i -row of A ℓ × ℓ = ( a ij ) ℓ × ℓ . By Lemma 2.5 , we know that the n um b er of the ℓ - tuples ( a 11 , . . . , a 1 ℓ ) ∈ F ℓ q \ { 0 } suc h that γ δk k + ℓ P i =1 a 2 1 i = 0 is N f  ℓ, γ δk + q − 1 2 k , q  , i.e., the v ector a 1 has N f  ℓ, γ δk + q − 1 2 k , q  9 c ho ices. Note that A ℓ × ℓ = ( a ij ) ℓ × ℓ ∈ GL ℓ ( F q ) if and o nly if b oth a i and a j are F q -linearly indep enden t for any 1 ≤ i 6 = j ≤ ℓ . No w for an y giv en v ector a 1 , t he total n umber of the v ector a 2 whic h is F q -linearly indep enden t o f a 1 is q ℓ − q . F urthermore, for giv en t wo ve ctors a 1 and a 2 , the total n um b er of t he v ector a 3 whic h is F q -linearly indep enden t o f b oth a 1 and a 2 is q ℓ − q 2 . In t he similar metho d a s the a b o v e, it is easy to know that for given i − 1 v ectors a 1 , . . . , a i − 1 , the tot al num b er of the v ector a i whic h is F q -linearly indep enden t of a 1 , . . . , a i − 1 is q ℓ − q i . And so, there are N f  ℓ, γ δk + q − 1 2 k , q  · ℓ − 1 Y i =1  q ℓ − q i  c ho ices fo r A ℓ × ℓ = ( a ij ) ℓ × ℓ that satisfies the condition (iii). In summary of the ab ov e discussions, the n um b er of G 1 ,k is less than or equal to  d  q − 1 2  − 1  · N f  ℓ, γ δk + q − 1 2 k , q  · ℓ − 1 Y i =1  q ℓ − q i  . F or t he case of ( a 11 , . . . , a 1 ℓ ) ∈  F ∗ q  ℓ , it’s easy to pro v e the corresp onding result via the similar pro ofs presen ted ab o v e. This completes the pro of o f Theorem 3 . 3 .  3.2 The second class of LCD GRL co des with the parameters [ k + 1 + ℓ, k ] In this subsection, b y taking α =  0 , γ δ α 1 , . . . , γ δ α k  , in the similar pro ofs as those for Theorems 3 . 1 - 3 . 2 , w e construct t w o classes of Euclidean LCD GRL co des, and then get tw o classes of GRL co des with 1-dimensional h ull and a class of GRL co des with 2-dimensional h ull, i.e., we prov e the follo wing Theorems 3 . 4 - 3.5 . Theorem 3.4 L et α =  0 , γ δ α 1 , . . . , γ δ α k  with gcd( k + 1 , q ) = 1 and 1 ≤ δ ≤ q − 1 . Then the fol lo w ing two statemen ts ar e true, (1) if ℓ < k 2 , then the c o de GRL k ( α , v , A ℓ × ℓ ) is Euclide an LCD; (2) if ℓ = k 2 and γ δk k + ℓ P i =1 a 2 1 i ∈ F ∗ q , then the c o de GRL k ( α , v , A ℓ × ℓ ) is Euclide an LCD. Remark 3.2 By taking δ = q − 1 and A l × l =  0 1 1 τ  with τ ∈ F q in Th e or em 3.4 ( 1 ), the c orr esp ondi ng r esult is just L emma 3.5 (2) in [ 36 ]. Theorem 3.5 L et 1 ≤ δ ≤ q − 1 an d α =  0 , γ δ α 1 , . . . , γ δ α k  with p | k + 1 . Then we have dim (Hull E (GRL k ( α , v , A ℓ × ℓ ))) =            1 , if ℓ < k 2 ; or ℓ = k 2 and γ δk k + ℓ P i =1 a 2 1 i ∈ F ∗ q ; 2 , if ℓ = k 2 and γ δk k + ℓ P i =1 a 2 1 i = 0 . 10 3.3 The third class of LCD GRL co des with the parameters [2 k + ℓ, k ] In this subsection, by taking α =  γ s α 1 , ..., γ s α k , γ t α 1 , ..., γ t α k  with 1 ≤ s 6 = t ≤ q − 1, w e construct t w o classes of Euclidean LCD G RL codes, a nd then get a class of GRL co des with 1-dimensional h ull. Firstly , w e presen t the follo wing key lemma. Lemma 3.1 I f q − 1 , k , s an d t satisfy q − 1 k ∤ s − t a n d v 2 ( s − t ) 6 = v 2 ( q − 1) − v 2 ( k ) − 1 , then any two c om p onents of the ve ctor α = ( γ s α 1 , ..., γ s α k , γ t α 1 , ..., γ t α k ) ar e distinct over F q and γ sk + γ tk ∈ F ∗ q . Pro of . Firstly , b y F ∗ q = h γ i , w e kno w that for a n y integers m and n , γ m = γ n if and only if m ≡ n ( mo d q − 1). And then for an y 1 ≤ i 6 = j ≤ k | q − 1, γ q − 1 k i 6 = γ q − 1 k j , i.e., α i 6 = α j . Note that γ ∈ F ∗ q , thus for a n y 1 ≤ i 6 = j ≤ k , w e hav e γ s α i 6 = γ s α j and γ t α i 6 = γ t α j . Hence, w e only need to pro v e that γ s α i 6 = γ t α j for an y s 6 = t and 1 ≤ i, j ≤ k . In fact, by α i 6 = α j (1 ≤ i ≤ k ) and α i = γ q − 1 k i (1 ≤ i ≤ k ), it’s easy to kno w that γ s α i 6 = γ t α j if and only if f o r any 1 ≤ i, j ≤ k , γ s α i 6 = γ t α j , i.e., for any 1 − k ≤ x ≤ k − 1, x q − 1 k 6≡ s − t ( mo d q − 1) . It’s w ell-kno wn that the binary linear Diophan tine equation x q − 1 k ≡ s − t ( mo d q − 1) is solv able on F q if and only if gcd  q − 1 k , q − 1  | s − t, i.e., q − 1 k | s − t . And so, an y tw o components of the ve ctor α = ( γ s α 1 , ..., γ s α k , γ t α 1 , ..., γ t α k ) with α i = γ q − 1 k i are distinct o v er F q if and only if q − 1 k ∤ s − t . Secondly , b y F ∗ q = h γ i , w e ha v e ord( γ ) = q − 1, namely , γ q − 1 = 1, it means γ q − 1 2 = − 1 . Note that γ sk + γ tk = 0 if a nd only if γ sk = − γ tk = γ q − 1 2 + tk , i.e., sk ≡ q − 1 2 + tk ( mo d q − 1) . F or the con v enience, we set r = q − 1 2 , i.e., q − 1 = 2 r . Th us sk ≡ q − 1 2 + tk ( mo d q − 1) if and only if there exists some w ∈ Z suc h tha t ( s − t ) k = (2 w + 1) r. It means v 2 (( s − t ) k ) = v 2 ((2 w + 1) r ) = v 2 ( r ) = v 2  q − 1 2  = v 2 ( q − 1) − 1 . Note that v 2 (( s − t ) k ) = v 2 ( s − t ) + v 2 ( k ), thus w e know t ha t if ( s − t ) k ≡ q − 1 2 ( mo d q − 1) , then v 2 ( s − t ) + 1 = v 2 ( q − 1) − v 2 ( k ) . And so, ( s − t ) k 6≡ q − 1 2 ( mo d q − 1) , i.e., γ sk + γ tk ∈ F ∗ q if v 2 ( s − t ) + 1 6 = v 2 ( q − 1) − v 2 ( k ). This completes the pro of o f Lemma 3 . 1 .  Esp ecially , when s = q − 1 and t = δ with 1 ≤ δ ≤ q − 2, i.e., α =  α 1 , ..., α k , γ δ α 1 , ..., γ δ α k  , w e can obta in the more precise result as the following 11 Lemma 3.2 I f q − 1 , k , δ satisfy o n e of the fol lowing c ond itions, then any two c om p onents of the ve ctor α =  α 1 , ..., α k , γ δ α 1 , ..., γ δ α k  ar e distinct over F q and 1 + γ δk ∈ F ∗ q . (1) v 2 ( k ) = v 2 ( q − 1 ) ≥ 1 and δ = 2 µ ( µ ≥ 1) ; (2) v 2 ( k ) < v 2 ( q − 1 ) and δ = 2 v 2 ( q − 1) − v 2 ( k ) − µ (1 < µ ≤ v 2 ( q − 1) − v 2 ( k )) ; (3) v 2 ( q − 1) − v 2 ( k ) 6 = 1 , and ther e exists an o dd prime p ′ such that v p ′ ( q − 1 ) = v p ′ ( k ) and δ = p v p ′ ( q − 1)+ µ i ( µ ≥ 1) ; (4) δ = 2 v 2 ( q − 1)+ µ ( µ ≥ 0 ) , and ther e exists an o dd prime p ′ such that v p ′ ( k ) < v p ′ ( q − 1 ) ; (5) v 2 ( q − 1) − v 2 ( k ) 6 = 1 , and ther e exists an o dd prime p ′ such that v p ′ ( k ) < v p ′ ( q − 1 ) and δ = ( p ′ ) v p ′ ( q − 1) − v p ′ ( k ) − µ (1 ≤ µ ≤ v p ′ ( q − 1) − v p ′ ( k )) . Based o n the ab ov e Lemmas 3 . 1 - 3 . 2 , in t he similar pro ofs as t hose f o r Theorems 3 . 1 - 3 .2 , one can obtain the following Theorems 3.6 - 3.8 . Theorem 3.6 L et gcd(2 k , q ) = 1 , α = ( γ s α 1 , ..., γ s α k , γ t α 1 , ..., γ t α k ) ∈ F 2 k q with q − 1 k ∤ s − t and v 2 ( s − t ) 6 = v 2 ( q − 1) − v 2 ( k ) − 1 . Then the c o d e G RL k ( α , v , A ℓ × ℓ ) is Euclide an LCD f o r ℓ < k 2 , or ℓ = k 2 and k  γ sk + γ tk  + ℓ P i =1 a 2 1 i ∈ F ∗ q . Theorem 3.7 L et gcd(2 k , q ) = 1 , α =  α 1 , ..., α k , γ δ α 1 , ..., γ δ α k  ∈ F 2 k q . If k , q − 1 , δ satisfy one of the fol low ing c ondition s, then the c o de GR L k ( α , v , A ℓ × ℓ ) is Euclide an LCD for ℓ < k 2 , or ℓ = k 2 and k  1 + γ δk  + ℓ P i =1 a 2 1 i ∈ F ∗ q . (1) v 2 ( k ) = v 2 ( q − 1 ) ≥ 1 and δ = 2 µ ( µ ≥ 1) ; (2) v 2 ( k ) < v 2 ( q − 1 ) and δ = 2 v 2 ( q − 1) − v 2 ( k ) − µ (1 < µ ≤ v 2 ( q − 1) − v 2 ( k )) ; (3) v 2 ( q − 1) − v 2 ( k ) 6 = 1 , ther e exists an o dd prime p ′ such that v p ′ ( q − 1) = v p ′ ( k ) , and δ = ( p ′ ) v p ′ ( q − 1)+ µ ( µ ≥ 1 ) ; (4) δ = 2 v 2 ( q − 1)+ µ ( µ ≥ 0 ) and ther e exists an o dd prime p ′ such that v p ′ ( k ) < v p ′ ( q − 1) ; (5) v 2 ( q − 1) − v 2 ( k ) 6 = 1 , ther e exists an o dd prime p ′ such that v p ′ ( k ) < v p ′ ( q − 1) , and δ = ( p ′ ) v p ′ ( q − 1) − v p ′ ( k ) − µ (1 ≤ µ ≤ v p ′ ( q − 1 ) − v p ′ ( k )) . Theorem 3.8 L et gcd (2 k , q ) = 1 and α =  α 1 , ..., α k , γ δ α 1 , ..., γ δ α k  ∈ F 2 k q . If k , q − 1 , δ satisfy on e of the fol lowing c onditions , ℓ = k 2 and k  1 + γ δk  + ℓ P i =1 a 2 1 i = 0 , then dim (Hull E (GRL k ( α , v , A ℓ × ℓ ))) = 1 . (1) v 2 ( k ) = v 2 ( q − 1 ) ≥ 1 and δ = 2 µ ( µ ≥ 1) ; (2) v 2 ( k ) < v 2 ( q − 1 ) and δ = 2 v 2 ( q − 1) − v 2 ( k ) − µ (1 < µ ≤ v 2 ( q − 1) − v 2 ( k )) ; (3) v 2 ( q − 1) − v 2 ( k ) 6 = 1 , ther e exists an o dd prime p ′ such that v p ′ ( q − 1) = v p ′ ( k ) , and δ = ( p ′ ) v p ′ ( q − 1)+ µ ( µ ≥ 1 ) ; (4) δ = 2 v 2 ( q − 1)+ µ ( µ ≥ 0 ) and ther e exists an o dd prime p ′ such that v p ′ ( k ) < v p ′ ( q − 1) ; (5) v 2 ( q − 1) − v 2 ( k ) 6 = 1 , ther e exists an o dd prime p ′ such that v p ′ ( k ) < v p ′ ( q − 1) , and δ = ( p ′ ) v p ′ ( q − 1) − v p ′ ( k ) − µ (1 ≤ µ ≤ v p ′ ( q − 1 ) − v p ′ ( k )) . 12 Remark 3.3 (1) By taking δ = 1 , µ = 0 and A l × l =  0 1 1 τ  with τ ∈ F q in The or em 3.7 (4), one c an imme di a tely get L em ma 3.5 (3) of the r efer enc e [ 36 ]. (2) It’s e asy to kn ow that in The or ems 3.6 - 3.8 , the c ondition gcd(2 k , q ) = 1 a l w ays holds when k | q − 1 and q is an o dd prime. Otherwise, if p | k , then by k | q − 1 , we have p | q − 1 , and so p | − 1 , w hich is a c ontr adi c tion . 3.4 Euclidean LCD GRL co des with the parameters [ 3 k + ℓ, k ] In this subsection, by taking α =  α 1 , ..., α k , γ α 1 , ..., γ α k , γ 2 α 1 , ..., γ 2 α k  , w e construct tw o classes of Euclidean LCD GRL co des, and then get three classes of GRL co des with 1-dimensional h ull and a class of G RL co des with 2-dimensional hull. Theorem 3.9 L et gcd(3 k , q ) = 1 , α = ( α 1 , ..., α k , γ α 1 , ..., γ α k , γ 2 α 1 , ..., γ 2 α k ) ∈ F 3 k q and q − 1 / ∈ { k, 2 k , 3 k } . Then the c o de GRL k ( α , v , A ℓ × ℓ ) is Euclide an LCD f o r ℓ < k 2 , or ℓ = k 2 and k  1 + γ k + γ 2 k  + ℓ P i =1 a 2 1 i ∈ F ∗ q . pro of. In t he similar pro of a s that for Theorem 3 . 7 , w e only need to pro v e that the follo wing t w o statemen ts are true, (1) for any 1 ≤ i 6 = j ≤ k | q − 1, α i 6 = γ α j , α i 6 = γ 2 α j and γ α i 6 = γ 2 α j ; (2) k  1 + γ k + γ 2 k  ∈ F ∗ q . F or (1). In the similar pro ofs as that for Lemma 3 . 1 , w e kno w that for any 1 ≤ i 6 = j ≤ k | q − 1, α i 6 = γ α j , α i 6 = γ 2 α j and γ α i 6 = γ 2 α j if a nd only if q − 1 k ∤ 1 and q − 1 k ∤ 2, i.e, the statemen t (1) holds if and only if q − 1 / ∈ { k , 2 k } . F or (2) . Note tha t 2 ≤ k | ord ( γ ) = q − 1, and so γ k − 1 6 = 0 if and only if k 6 = q − 1 . By  1 + γ k + γ 2 k   γ k − 1  = γ 3 k − 1, we kno w that 1 + γ k + γ 2 k ∈ F ∗ q if a nd only if γ 3 k − 1 ∈ F ∗ q and k 6 = q − 1, i.e., ord ( γ ) = q − 1 ∤ 3 k and k 6 = q − 1, namely , q − 1 k ∤ 3 and k 6 = q − 1, it means that the statemen t (2) holds if and only if q − 1 / ∈ { k , 3 k } . F rom the ab ov e discussions, Theorem 3 . 9 is immediately .  In the similar analysis as that for Remark 3.3 (2), it ’s easy to kno w that if p 6 = 3 a nd k | q − 1, then gcd( 3 k , q ) = 1, and so w e can get the follo wing Theorem 3.10 L et p ≥ 5 , α = ( α 1 , ..., α k , γ α 1 , ..., γ α k , γ 2 α 1 , ..., γ 2 α k ) ∈ F 3 k q and q − 1 / ∈ { k , 2 k , 3 k } . Then the c o de G RL k ( α , v , A ℓ × ℓ ) is Euclide an LCD for ℓ < k 2 , or ℓ = k 2 and k  1 + γ k + γ 2 k  + ℓ P i =1 a 2 1 i ∈ F ∗ q . In the similar pro of as that for Theorem 3.2 , one can obtain the following 13 Theorem 3.11 L et α = ( α 1 , ..., α k , γ α 1 , ..., γ α k , γ 2 α 1 , ..., γ 2 α k ) ∈ F 3 k q . Then for q − 1 / ∈ { k , 2 k , 3 k } , we hav e dim (Hull E (GRL k ( α , v , A ℓ × ℓ ))) =                    1 , if p = 3 , ℓ < k 2 , or p = 3 , ℓ = k 2 and k  1 + γ k + γ 2 k  + ℓ P i =1 a 2 1 i ∈ F ∗ q , or gcd(3 k , q ) = 1 , ℓ = k 2 and k  1 + γ k + γ 2 k  + ℓ P i =1 a 2 1 i = 0; 2 , if p = 3 , ℓ = k 2 and k  1 + γ k + γ 2 k  + ℓ P i =1 a 2 1 i = 0 . 3.5 Sev eral classes of EA QECCs In this subsection, combinin g Theorems 3.1 - 3.9 and Lemma 2.8 , w e can immediately obtain sev eral classes o f EA QECCs as the fo llo wing Theorem 3.12 Assume that d is the mini m um distanc e fo r the c o de GRL k ( α , v , A ℓ × ℓ ) . Then ther e e xists some q -ary EAQECCs with o n e of the fol lowing p ar am e ters, (1) [[ k + ℓ, k − i, d, ℓ − i ]] q for i = 0 , 1 ; (2) [[ k + 1 + ℓ , k − i, d, ℓ + 1 − i ]] q for i = 0 , 1 , 2 ; (3) [[2 k + ℓ, k − i, d, k + ℓ − i ]] q for i = 0 , 1 ; (4) [[3 k + ℓ, k − i, d, 2 k + ℓ − i ]] q for i = 0 , 1 , 2 . In fact, f o r the giv en GRL co de GRL k ( α , v , A ℓ × ℓ ), its Euclide an dua l co de is unique ly determined. F urthermore, w e also can immediately obtain sev eral classes of ESQECCs as the follo wing Theorem 3.13 Assume that d ⊥ E is the minimum distanc e for the c o de GRL ⊥ E k ( α , v , A ℓ × ℓ ) . Then ther e exists some q -ary EAQECCs with one of the fol lowi n g p ar ameters, (1)  k + ℓ, ℓ − i, d ⊥ E , k − i  q for i = 0 , 1 ; (2)  k + 1 + ℓ , ℓ − i, d ⊥ E , k + 1 − i  q for i = 0 , 1 , 2 ; (3)  2 k + ℓ, k + ℓ − i, d ⊥ E , k − i  q for i = 0 , 1 ; (4)  3 k + ℓ, 2 k + ℓ − i, d ⊥ E , k − i  q for i = 0 , 1 , 2 . 4 Hermitian LCD GRL co de s and th eir applic ati ons Throughout this section, w e fix k | q 2 − 1, F ∗ q 2 = F q 2 \ { 0 } = h γ i and α i = γ q 2 − 1 k i (1 ≤ i ≤ k ). In this section, b y taking some sp ecial v ector α = ( α 1 , . . . , α n ), w e construct four classes of Hermitian LCD GRL co des with the parameters [ n + ℓ, k ], get sev eral classes of GRL co des with small-dimensional h ulls, and then f o r some G RL co des, w e obtain an upp er b ound for the dimension of the h ull. Finally , w e obtain sev eral families of EA QECCs. Firstly , by Lemma 2.3 , it’s easy to obtain the follow ing crucial lemma. 14 Lemma 4.1 L et α = ( γ t α 1 , . . . , γ t α k ) with 1 ≤ t ≤ q 2 − 1 , α i = γ q 2 − 1 k i and α i 6 = α j for 1 ≤ i 6 = j ≤ k . Then ther e exis ts exactly one non -zer o element over F q 2 for e ach r ow or e ach c olumn of the matrix M k ,t =            k k P i =1 ( γ t α i ) q k P i =1 ( γ t α i ) 2 q · · · k P i =1 ( γ t α i ) ( k − 1) q k P i =1 ( γ t α i ) k P i =1 ( γ t α i ) 1+ q k P i =1 ( γ t α i ) 1+2 q · · · k P i =1 ( γ t α i ) 1+( k − 1) q . . . . . . . . . . . . k P i =1 ( γ t α i ) k − 1 k P i =1 ( γ t α i ) k − 1+ q k P i =1 ( γ t α i ) k − 1+2 q · · · k P i =1 ( γ t α i ) k − 1+( k − 1) q            . Pro of. F or any giv en s (1 ≤ s ≤ k ), the s -column of M k ,t is            k P i =1 ( γ t α i ) 0+( s − 1) q k P i =1 ( γ t α i ) 1+( s − 1) q . . . k P i =1 ( γ t α i ) k − 1+( s − 1) q            . Note that { 0 + sq , 1 + sq , · · · , k − 1 + sq } mod k = { 0 , 1 , . . . , k − 1 } , th us b y Lemma 2.3 , ev ery column of M k ,t has exactly one non-zero elemen t. No w, for a n y give n s (1 ≤ s ≤ k ), the s -row of M k ,t is  k P i =1 ( γ t α i ) ( s − 1)+0 · q k P i =1 ( γ t α i ) ( s − 1)+1 · q · · · k P i =1 ( γ t α i ) ( s − 1)+( k − 1) · q  . Note that ( s − 1) + 0 · q , ( s − 1) + 1 · q , · · · , ( s − 1) + ( k − 1) q , are k p ositiv e in tegers, a nd for 0 ≤ i 6 = j ≤ k − 1, b y k | q 2 − 1 and q = p m ( m ∈ Z + ), we hav e gcd( p, k ) = 1 and 1 − k ≤ i − j ≤ k − 1, furthermore, it’s easy to know that ( s − 1) + iq ≡ ( s − 1) + j q ( mo d k ) ⇐ ⇒ ( i − j ) q ≡ 0 ( mo d k ) ⇐ ⇒ k | ( i − j ) q ⇐ ⇒ k | ( i − j ) , whic h is a con tradiction. And so, for 0 ≤ i 6 = j ≤ k − 1, w e ha v e ( s − 1) + iq 6≡ ( s − 1) + j q ( mo d k ) , it means tha t { ( s − 1) + 0 · q , ( s − 1) + 1 · q , · · · , ( s − 1) + ( k − 1) q } mod k = { 0 , 1 , . . . , k − 1 } , th us b y Lemma 2.3 , ev ery row of M k ,t has exactly one non- zero elemen t. F rom the ab ov e discussions, w e complete the pro of of Lemma 4 . 1 .  Esp ecially , b y taking k | q − 1 or k | q + 1 in L emma 4.1 , w e can obtain the follo wing 15 Corollary 4.1 L et α = ( γ t α 1 , . . . , γ t α k ) with 1 ≤ t ≤ q 2 − 1 . The n the fol low ing two state- ments ar e true, (1) if k | q − 1 , then the ( i, k − i ) -th entry is non-zer o of M k ,t for 1 ≤ i ≤ k − 1 , i.e., M k ,t =        k 0 0 · · · 0 0 0 0 · · · ( γ t ) 1+( k − 1) q k . . . . . . . . . . . . 0 0 ( γ t ) k − 2+2 q k · · · 0 0 ( γ t ) k − 1+ q k 0 · · · 0        ; (2) if k | q + 1 , then the ( i, i ) -th entry is non-zer o of M k ,t for 2 ≤ i ≤ k , i.e., M k ,t =        k 0 · · · 0 0 0 ( γ t ) 1+ q k · · · 0 0 . . . . . . . . . . . . 0 0 0 · · · ( γ t ) k − 2+( k − 2) q k 0 0 0 · · · 0 ( γ t ) k − 1+( k − 1) q k        . 4.1 Hermitian L CD GRL co des with the parameters [ k + ℓ, k ] In this subsection, by ta king α =  γ δ α 1 , . . . , γ δ α k  with 1 ≤ δ ≤ q 2 − 1, w e construct tw o classes of Hermitian LCD G RL co des, and t hen get a class of GRL co des with 1-dimensional h ull. And for a class of GRL co des, w e o bt a in an upp er b ound for the dimension of the h ull. Theorem 4.1 L et α =  γ δ α 1 , . . . , γ δ α k  with k | q − 1 and 1 ≤ δ ≤ q 2 − 1 . Then the fol low i n g two statemen ts ar e true, (1) if ℓ < k 2 , then the c o de GRL k ( α , v , A ℓ × ℓ ) is Hermitian LCD; (2) if ℓ = k 2 and k γ δ ( k − ℓ + ℓq ) + ℓ P i =1 a 1+ q 1 i ∈ F ∗ q 2 , then the c o de GR L k ( α , v , A ℓ × ℓ ) is Hermitian LCD. Pro of . By Remark 2.1 , w e only fo cus on G RL k ( α , 1 , A l × l ) with the generator matrix G 1 ,k giv en b y ( 2 . 2 ). F urthermore, b y Lemma 2.2 , w e only need to prov e tha t the k × k ma- trix G 1 ,k G T 1 ,k is nonsingular ov er F q 2 , i.e., rank  G 1 ,k  G 1 ,k  T  = k . In fact, let L k × ℓ =  0 ( k − ℓ ) × ℓ A ℓ × ℓ  , then G 1 ,k  G 1 ,k  T = M k ,δ + L ℓ  L ℓ  T . 16 No w by Corolla ry 4.1 , we ha v e G 1  G 1  T =                                k 0 · · · 0 0 · · · 0 0 · · · 0 0 · · · 0 0 0 · · · 0 0 · · · 0 0 · · · 0 0 · · · Υ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · 0 0 · · · 0 0 · · · 0 Υ ℓ · · · 0 0 0 · · · 0 0 · · · 0 0 · · · Υ ℓ +1 0 · · · 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · 0 0 · · · 0 . . . · · · 0 0 · · · 0 0 0 · · · 0 0 · · · . . . 0 · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · 0 Υ k − ℓ − 1 · · · 0 0 · · · 0 0 · · · 0 0 0 · · · Υ k − ℓ 0 · · · 0 0 · · · 0 ℓ P i =1 a 1+ q 1 i · · · ℓ P i =1 a 1 i a q ℓi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 Υ k − 1 · · · 0 0 · · · 0 0 · · · 0 ℓ P i =1 a ℓi a q 1 i · · · ℓ P i =1 a 1+ q ℓi                                , where Υ i = k γ δ ( i +( k − i ) q ) (1 ≤ i ≤ k − 1). Note that 2 ≤ k | q − 1 and F ∗ q 2 = F q 2 \ { 0 } = h γ i , thus w e hav e k , Υ i ∈ F ∗ q 2 (1 ≤ i ≤ k − 1). F ur t hermore, by Lemma 2.2 , the co de GRL k ( α , v , A ℓ × ℓ ) is Hermitian LCD when ℓ < k 2 , or ℓ = k 2 and k γ δ ( k − ℓ + ℓq ) + ℓ P i =1 a 1+ q 1 i ∈ F ∗ q 2 . F rom the ab ov e discussions, w e complete the pro of of Theorem 4 . 1 .  Remark 4.1 By taking δ = q 2 − 1 an d A l × l =  0 1 1 τ  with τ ∈ F 2 q in T he or em 4 .1 (1 ) , on e c an get L emma 3.12 ( 1 ) of the r efer enc e [ 36 ]. By Lemma 2.7 and Corollary 4.1 (2), it’s easy to pr ov e that the follow ing t wo theorems. Theorem 4.2 L et α =  γ δ α 1 , . . . , γ δ α k  with k | q + 1 and 1 ≤ δ ≤ q 2 − 1 . Then dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) ≤ ℓ. Remark 4.2 By taking ( q , k , ℓ, δ ) = (9 , 5 , 3 , 1) a n d A ℓ × ℓ =   γ 2 0 0 0 γ 3 0 0 0 γ 4   in The or em 4.3 , and b asi n g on the Magma pr o gr am , we ha v e dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) = 3 , which me ans that the b ound in The or em 4.3 is attainable. Theorem 4.3 L et k | q − 1 , α =  γ δ α 1 , . . . , γ δ α k  with 1 ≤ δ ≤ q 2 − 1 . If ℓ = k 2 and k γ δ ( k − ℓ + ℓq ) + ℓ P i =1 a 1+ q 1 i = 0 , then dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) = 1 . 17 4.2 Hermitian L CD GRL co des with the parameters [ k + 1 + ℓ, k ] In this subsection, by taking α =  0 , γ δ α 1 , . . . , γ δ α k  with 1 ≤ δ ≤ q 2 − 1 , in the similar pro ofs as those for Theorems 4.1 - 4.3 , w e construct t w o classes of Hermitian LCD GR L co des, and then get t w o classes of GR L co des with 1 -dimensional hull and a class o f GRL co des with 2-dimensional h ull. And for a class of GRL co des, w e obtain an upp er b ound for the dimension of the hull as the follow ing Theorems 4.4 - 4.6 . Theorem 4.4 L et k | q − 1 , gcd( k + 1 , q ) = 1 and α =  0 , γ δ α 1 , . . . , γ δ α k  with 1 ≤ δ ≤ q 2 − 1 . then the fol lowi n g two statements ar e true, (1) if ℓ < k 2 , then the c o de GRL k ( α , v , A ℓ × ℓ ) is Hermitian LCD; (2) if ℓ = k 2 and k γ δ ( k − ℓ + ℓq ) + ℓ P i =1 a 1+ q 1 i ∈ F ∗ q 2 , then the c o de GR L k ( α , v , A ℓ × ℓ ) is Hermitian LCD. Remark 4.3 By taking q 2 − 1 | δ a n d A l × l =  0 1 1 τ  with τ ∈ F 2 q in Th e or em 4.4 ( 1 ) , one c an get L emma 3.12 ( 2 ) of the r efer enc e [ 36 ]. Theorem 4.5 L et p | k + 1 and α =  0 , γ δ α 1 , . . . , γ δ α k  with 1 ≤ δ ≤ q 2 − 1 . Then we have dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) =            1 , if ℓ < k 2 ; or ℓ = k 2 and k γ δ ( k − ℓ + ℓq ) + ℓ P i =1 a 1+ q 1 i ∈ F ∗ q 2 ; 2 , if ℓ = k 2 and k γ δ ( k − ℓ + ℓq ) + ℓ P i =1 a 1+ q 1 i = 0 . Theorem 4.6 L et α =  0 , γ δ α 1 , . . . , γ δ α k  with ( k + 1 , q ) = 1 , k | q + 1 and 1 ≤ δ ≤ q 2 − 1 . Then dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) ≤ ℓ. Remark 4.4 By taking ( q , k , ℓ, δ ) = (11 , 6 , 3 , 1) and A ℓ × ℓ =   γ 4 0 0 0 γ 5 0 0 0 γ 6   in The or e m 4.6 , and b asi n g on the Magma pr o gr am , we ha v e dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) = 3 , which me ans that the b ound in The or em 4.6 is attainable. 4.3 Hermitian L CD GRL co des with the parameters [ 2 k + ℓ, k ] In this subsection, by taking α =  γ s α 1 , ..., γ s α k , γ t α 1 , ..., γ t α k  with 1 ≤ s 6 = t ≤ q 2 − 1, in t he similar pro of as that for Theorem 3.6 , we construct tw o classes of Hermitain LCD GRL co des, and for a class of GRL co des, w e obtain a n upp er b ound for the dimension of the h ull . Firstly , in the similar pro ofs as those for Lemmas 3.1 - 3.2 , we can obtain the following key lemmas and corolla rys. 18 Lemma 4.2 Any two c o m p onents of the v e ctor α = ( γ s α 1 , ..., γ s α k , γ t α 1 , ..., γ t α k ) with 1 ≤ s 6 = t ≤ q 2 − 1 ar e distinct if and only if q 2 − 1 k ∤ s − t. Corollary 4.2 Any two c omp on e nts of the ve ctor α =  α 1 , ..., α k , γ δ α 1 , ..., γ δ α k  with 1 ≤ δ ≤ q 2 − 1 ar e distinct if and only if q 2 − 1 k ∤ δ. Lemma 4.3 L et T =  v 2 ( q 2 − 1) − 1 − v 2 ( i + ( k − i ) q ) | 1 ≤ i ≤ k − 1  . Then for any 1 ≤ i ≤ k − 1 , γ s ( i +( k − i ) q ) + γ t ( i +( k − i ) q ) ∈ F ∗ q 2 if and only if ( s − t ) ( i + ( k − i ) q ) 6≡ q 2 − 1 2 ( mo d q 2 − 1) . Esp e ci a l ly, for v 2 ( s − t ) / ∈ T , w e have γ s ( i +( k − i ) q ) + γ t ( i +( k − i ) q ) ∈ F ∗ q 2 for a n y 1 ≤ i ≤ k − 1 . Corollary 4.3 L et T =  v 2 ( q 2 − 1) − 1 − v 2 ( i + ( k − i ) q ) | 1 ≤ i ≤ k − 1  . Then for any 1 ≤ i ≤ k − 1 , 1 + γ δ ( i +( k − i ) q ) ∈ F ∗ q 2 if and only if δ ( i + ( k − i ) q ) 6≡ q 2 − 1 2 ( mo d q 2 − 1) . Esp e cial ly, for v 2 ( δ ) / ∈ T , we have 1 + γ δ ( i +( k − i ) q ) ∈ F ∗ q 2 for a n y 1 ≤ i ≤ k − 1 . Basing on the a b o v e Lemmas 4 . 2 - 4 . 3 , w e can obtain the follo wing Theorems 4.7 - 4.8 . Theorem 4.7 L et T =  v 2 ( q 2 − 1) − 1 − v 2 ( i + ( k − i ) q ) | 1 ≤ i ≤ k − 1  , and α = ( γ s α 1 , ..., γ s α k , γ t α 1 , ..., γ t α k ) ∈ F 2 k q 2 with 1 ≤ s 6 = t ≤ q 2 − 1 . If v 2 ( s − t ) / ∈ T , q 2 − 1 k ∤ s − t, gcd(2 k , q ) = 1 and k | q − 1 , then the c o de GRL k ( α , v , A ℓ × ℓ ) is Hermitian LC D for ℓ < k 2 or ℓ = k 2 and k  γ s ( k − ℓ + ℓq ) + γ t ( k − ℓ + ℓq )  + ℓ P i =1 a 1+ q 1 i ∈ F ∗ q 2 . Theorem 4.8 L et T =  v 2 ( q 2 − 1) − 1 − v 2 ( i + ( k − i ) q ) | 1 ≤ i ≤ k − 1  , and α =  α 1 , ..., α k , γ δ α 1 , ..., γ δ α k  ∈ F 2 k q 2 with 1 ≤ δ ≤ q 2 − 1 . If v 2 ( δ ) / ∈ T , q 2 − 1 k ∤ δ, gcd(2 k , q ) = 1 and k | q − 1 , then the c o de GRL k ( α , v , A ℓ × ℓ ) is Hermitian LCD for ℓ < k 2 or ℓ = k 2 and k  1 + γ δ ( k − ℓ + ℓq )  + ℓ P i =1 a 1+ q 1 i ∈ F ∗ q 2 . Esp ecially , when δ = 2 v 2 ( q 2 − 1 ) + µ ( µ ≥ 0), it’s easy to get v 2 ( δ ) / ∈ T . F urthermore, w e can obtain the fo llowing Theorem 4.9 L et α =  α 1 , ..., α k , γ δ α 1 , ..., γ δ α k  ∈ F 2 k q 2 with 1 ≤ δ ≤ q 2 − 1 and δ = 2 v 2 ( q 2 − 1 ) + µ ( µ ≥ 0) . If gcd(2 k , q ) = 1 , k | q − 1 , and ther e e x i s ts an o dd prime p ′ such that v p ′ ( k ) < v p ′ ( q 2 − 1) , then the c o de G RL k ( α , v , A ℓ × ℓ ) is Hermitian LCD for ℓ < k 2 or ℓ = k 2 and k  1 + γ δ ( k − ℓ + ℓq )  + ℓ P i =1 a 1+ q 1 i ∈ F ∗ q 2 . 19 Remark 4.5 By taking µ = 0 and A l × l =  0 1 1 τ  with τ ∈ F 2 q in The or em 4.9 , the c orr e- sp onding r esult is just L emma 3.12 (3) in [ 36 ]. Similarly , if there exists an o dd prime p ′ suc h that v p ′ ( k ) = v p ′ ( q 2 − 1) and δ = ( p ′ ) µ ( µ ≥ 1), w e hav e v 2 ( δ ) = 0. F urthermore, w e can obtain the followin g Theorem 4.10 L et α =  α 1 , ..., α k , γ δ α 1 , ..., γ δ α k  ∈ F 2 k q 2 with 1 ≤ δ ≤ q 2 − 1 . If g cd(2 k , q ) = 1 , k | q − 1 , 0 / ∈ T , and ther e exi s ts an o dd prime p ′ such that v p ′ ( k ) = v p ′ ( q 2 − 1) and δ = ( p ′ ) µ ( µ ≥ 1) , then the c o de GRL k ( α , v , A ℓ × ℓ ) is Hermitian LCD f or ℓ < k 2 or ℓ = k 2 and k  1 + γ δ ( k − ℓ + ℓq )  + ℓ P i =1 a 1+ q 1 i ∈ F ∗ q 2 . In the similar pro ofs as those for Theorem 4.3 and Theorem 4.7 , one can obt a in the follo wing Theorem 4.11 L et N ℓ = { v 2 ( q − 1) − v 2 ( i ) − 1 | 1 ≤ i ≤ k − ℓ − 1 } , and α = ( γ s α 1 , ..., γ s α k , γ t α 1 , ..., γ t α k ) ∈ F 2 k q 2 with k | q + 1 and 1 ≤ s 6 = t ≤ q 2 − 1 . If v 2 ( s − t ) / ∈ N ℓ and q 2 − 1 k ∤ s − t , then dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) ≤ ℓ. Remark 4.6 By taking ( q , k , ℓ, s, t ) = (3 2 , 5 , 3 , 9 , 1) a n d A ℓ × ℓ =   γ 6 0 0 0 γ 7 0 0 0 γ 8   in The or em 4.11 , and b asing on the Magm a pr o gr a m , we ha ve dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) = 3 , which me ans that the b ound in The or em 4.11 is attainable. 4.4 Hermitian L CD G RL co d es with the parameters [( δ + 1) k + ℓ, k ] In this subsection, by taking α =  α 1 , ..., α k , γ α 1 , ..., γ α k , . . . , γ δ α 1 , ..., γ δ α k  with 1 ≤ δ ≤ q , we construc t tw o classes of Hermitian LCD GRL co des, and then get t w o classes of GRL co des with 1-dimensional h ull and a class of GRL co des with 2-dimensional h ull. And for a class of GRL co des, w e o bt a in an upp er b ound for the dimension of the h ull. Theorem 4.12 L et ∆ i = i + ( k − i ) q (1 ≤ i ≤ k − 1) and α =  α 1 , ..., α k , γ α 1 , ..., γ α k , . . . , γ δ α 1 , ..., γ δ α k  with 1 ≤ δ ≤ q . If gcd (( δ + 1) k , q ) = 1 , k | q − 1 , and q 2 − 1 gcd( q 2 − 1 , ∆ i ) ∤ δ + 1 for a n y 1 ≤ i ≤ k − 1 . Th en the c o de GRL k ( α , v , A ℓ × ℓ ) is Hermitian LCD for ℓ < k 2 or ℓ = k 2 and k δ P j = 0 γ j ( ℓ +( k − ℓ ) q ) + ℓ P i =1 a 1+ q 1 i ∈ F ∗ q 2 . 20 Pro of . In the similar pro of as that for Theorem 4 . 1 , w e only need to prov e that the follo wing t w o statemen ts are true, (1) for any 1 ≤ i, j ≤ k a nd 0 ≤ s, t ≤ δ , γ s α i 6 = γ t α j ; (2) for any 1 ≤ i ≤ k − 1, δ P j = 0 γ j ( i +( k − i ) q ) ∈ F ∗ q 2 . F or (1). In the similar pro of a s t hat for Lemma 3 .1 , w e kno w that the statemen ts ( 1) holds if and only if q 2 − 1 k ∤ t for an y − δ ≤ t ≤ δ . By q + 1 | q 2 − 1 and k | q − 1, w e kno w that if there exists some − δ ≤ t ≤ δ suc h that q 2 − 1 k | t , then q + 1 | t , which is contradict with t ≤ δ ≤ q . F or (2). By k | q − 1 , 1 ≤ i ≤ k − 1 and ∆ i = k q + (1 − q ) i , we ha v e − ( q 2 − 4 q + 2) ≤ ∆ i ≤ q 2 − 2 q + 1 . Since q is an o dd prime p ow er, thus for an y 1 ≤ i ≤ k − 1, − ( q 2 − 1 ) < ∆ i < q 2 − 1 , i.e., γ ∆ i − 1 6 = 0 for an y 1 ≤ i ≤ k − 1. Note t ha t  γ ∆ i − 1  δ X j = 0 γ j ∆ i = γ ( δ +1)∆ i − 1 , then δ P j = 0 γ j ∆ i ∈ F ∗ q 2 if a nd only if γ ( δ +1)∆ i − 1 ∈ F ∗ q 2 , i.e., ord ( γ ) = q 2 − 1 ∤ ( δ + 1)∆ i , namely , q 2 − 1 gcd ( q 2 − 1 , ∆ i ) ∤ δ + 1 . F rom the ab ov e discussions, w e complete the pro of of Theorem 4 . 12 .  In the similar pro of as that for Theorem 3.2 , one can obtain the following Theorem 4.13 L et ∆ i = i + ( k − i ) q (1 ≤ i ≤ k − 1) , S t =  i     q 2 − 1 gcd ( q 2 − 1 , ∆ i ) | δ + 1 , t ≤ i ≤ k − 1  and α =  α 1 , ..., α k , γ α 1 , ..., γ α k , . . . , γ δ α 1 , ..., γ δ α k  with 1 ≤ δ ≤ q , k | q − 1 . T h en we h ave dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) =                              1 , if # S 1 = 0 , p | δ + 1 and ℓ < k 2 ; or # S 1 = 0 , p | δ + 1 , ℓ = k 2 , and k δ P i =0 γ δ ( ℓ +( k − ℓ ) q ) + ℓ P i =1 a 1+ q 1 i ∈ F ∗ q 2 ; or # S 1 = 0 , g cd ( k ( δ + 1) , q ) = 1 , ℓ = k 2 , a nd k δ P i =0 γ δ ( ℓ +( k − ℓ ) q ) + ℓ P i =1 a 1+ q 1 i = 0; 2 , if # S 1 = 0 , p | δ + 1 , ℓ = k 2 , and k δ P i =0 γ δ ( ℓ +( k − ℓ ) q ) + ℓ P i =1 a 1+ q 1 i = 0; s, if # S k − ℓ = s, gcd ( k ( δ + 1) , q ) = 1 , ℓ < k 2 ; s + 1 , if # S k − ℓ = s, p | δ + 1 , ℓ < k 2 . 21 Theorem 4.14 L et α =  α 1 , ..., α k , γ α 1 , ..., γ α k , . . . , γ δ α 1 , ..., γ δ α k  with 1 ≤ δ ≤ q , k | q + 1 , Λ i = i ( q + 1)(1 ≤ i ≤ k − 1) an d U ℓ =  i     q 2 − 1 gcd ( q 2 − 1 , Λ i ) | δ + 1 , 1 ≤ i ≤ k − ℓ − 1  . If gcd (( δ + 1) k , q ) = 1 an d # U ℓ = 0 , then dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) ≤ ℓ. Remark 4.7 By taking ( q , k , ℓ, δ ) = (3 2 , 5 , 3 , 3) an d A ℓ × ℓ =   γ 8 0 0 0 γ 5 0 0 0 γ 8   in T he or em 4.14 , and b asi n g on the Magma pr o gr am , we ha v e dim (Hull H (GRL k ( α , v , A ℓ × ℓ ))) = 3 , which me ans that the b ound in The or em 4.14 is attainable. 4.5 Sev eral classes of EA QECCs In this subs ection, com bining Lemma 2.10 a nd Theorems 4.1 , 4.3 - 4.5 , 4.7 - 4 .10 , 4 .12 - 4.13 , w e can immediately obtain sev eral classes of ESQECCs as the f o llo wing Theorem 4.15 Assume that d is the mini m um distanc e fo r the c o de GRL k ( α , v , A ℓ × ℓ ) . Then ther e e xists some q-ary EAQECCs with one o f the fol lowin g p ar am eters, (1) [[ k + ℓ, k − i, d, ℓ − i ]] q for i = 0 , 1 ; (2) [[ k + 1 + ℓ , k − i, d, ℓ + 1 − i ]] q for i = 0 , 1 , 2 ; (3) [[2 k + ℓ, k , d, k + ℓ ]] q ; (4) [[( δ + 1) k + ℓ, k − i, d, δ k + ℓ − i ]] q for i = 0 , 1 , 2 , . . . , s, s + 1 with s ≤ ℓ . In fact, for the given G R L co de GRL k ( α , v , A ℓ × ℓ ), its Hermitian dua l co de is uniquely determined. F urthermore, w e also can immediately obtain sev eral classes of ESQECCs as the follo wing Theorem 4.16 Assume that d ⊥ H is the minimum distanc e for the c o de GRL ⊥ H k ( α , v , A ℓ × ℓ ) . Then ther e exists some q-a ry EAQECCs with one of the fol lowing p a r ameters, (1)  k + ℓ, ℓ − i, d ⊥ H , k − i  q for i = 0 , 1 ; (2)  k + 1 + ℓ , ℓ + 1 − i, d ⊥ H , k − i  q for i = 0 , 1 , 2 ; (3)  2 k + ℓ, k + ℓ, d ⊥ H , k  q ; (4)  ( δ + 1) k + ℓ, δ k + ℓ − i, d ⊥ H , k − i  q for i = 0 , 1 , 2 , . . . , s, s + 1 w i th s ≤ ℓ . 5 The non- RS prop ert y of the GRL co de In this section, by using the Cauc h y matrix metho d prop osed in [ 32 ], w e study the non- GRS prop ert y of t he co de GRL k ( α , v , A ℓ × ℓ ), and prov e that the co de GRL k ( α , v , A ℓ × ℓ ) is non-GRS for k > ℓ . And when k = ℓ , w e g iv e some examples. Firstly , w e presen t the follo wing crucial lemma. 22 Lemma 5.1 L et σ i b e the i -th de gr e e elementary symmetric p olynom ial and f ( x ) = k Y i =1 ( x − α i ) = k X i =0 ( − 1) i σ i x k − i . Then for f i ( x ) = k X j = 1 f ij x j − 1 = k Y j = 1 ,j 6 = i ( x − α j ) (1 ≤ i ≤ k ) , we have f ij = k − j X s =0 ( − 1) s σ s α k − j − s i (1 ≤ j ≤ k ) . Pro of . Note that f ( x ) = ( x − α i ) f i ( x ) , then b y comparing the co efficien ts for x i in b oth sides, w e ha v e                  f ik = 1 , f i ( k − 1) − f ik α i = − σ 1 , f i ( k − 2) − f i ( k − 1) α i = σ 2 , . . . f i 1 − f i 2 α i = ( − 1) k − 1 σ k − 1 , − f i 1 α i = ( − 1) k σ k . F urthermore, w e can get                          f ik = 1 , f i ( k − 1) = α i − σ 1 , f i ( k − 2) = ( α i − σ 1 ) α i + σ 2 , . . . f i 2 = k − 2 P s =0 ( − 1) s σ s α k − 2 − s i , f i 1 = k − 1 P s =0 ( − 1) s σ s α k − 1 − s i . This completes the pro of o f Lemma 5 . 1 .  No w, w e prov e that the co de G RL k ( α , v , A ℓ × ℓ ) is non-G RS when k > ℓ . Theorem 5.1 I f k > ℓ , then the c o d e GRL k ( α , v , A ℓ × ℓ ) is non-GRS. Pro of . By Remark 2.1 , w e only fo cus on the co de GRL k ( α , 1 , A ℓ × ℓ ). Firstly , we set f i ( x ) = k X j = 1 f ij x j − 1 = k Y j = 1 ,j 6 = i ( x − α j ) (1 ≤ i ≤ k ) , 23 F =     f 11 f 12 · · · f 1 k f 21 f 22 · · · f 2 k . . . . . . . . . f k 1 f k 2 · · · f k k     (5.1) and η i = k Y s =1 ,s 6 = i ( α i − α s ) (1 ≤ i ≤ k ) , η k + j = k Y s =1 ( α k + j − α s ) (1 ≤ j ≤ n − k ) . No w for F and G 1 giv en b y ( 5 . 1 ) and ( 2 . 2 ) , resp ectiv ely , w e ha v e F G 1 =        f 1 ( α 1 ) · · · f 1 ( α k ) f 1 ( α k +1 ) · · · f 1 ( α n ) ℓ P s =1 f 1( k − ℓ + s ) a s 1 · · · ℓ P s =1 f 1( k − ℓ + s ) a sℓ . . . . . . . . . . . . . . . . . . f k ( α 1 ) · · · f k ( α k ) f k ( α k +1 ) · · · f k ( α n ) ℓ P s =1 f k ( k − ℓ + s ) a s 1 · · · ℓ P s =1 f k ( k − ℓ + s ) a sℓ        . Note that f i ( α j ) =      η j , 1 ≤ i = j ≤ k ; 0 , 1 ≤ i 6 = j ≤ k ; η j α j − α i , k + 1 ≤ j ≤ n, th us F G 1 =        η 1 · · · 0 η k +1 α k +1 − α 1 · · · η n α n − α 1 ℓ P s =1 f 1( k − ℓ + s ) a s 1 · · · ℓ P s =1 f 1( k − ℓ + s ) a sℓ . . . . . . . . . . . . . . . . . . 0 · · · η k η k +1 α k +1 − α k · · · η n α n − α k ℓ P s =1 f k ( k − ℓ + s ) a s 1 · · · ℓ P s =1 f k ( k − ℓ + s ) a sℓ        =   η 1 · · · 0 . . . . . . 0 · · · η k          1 · · · 0 η k +1 η − 1 1 α k +1 − α 1 · · · η n η − 1 1 α n − α 1 η − 1 1 ℓ P s =1 f 1( k − ℓ + s ) a s 1 · · · η − 1 1 ℓ P s =1 f 1( k − ℓ + s ) a sℓ . . . . . . . . . . . . . . . . . . 0 · · · 1 η k +1 η − 1 k α k +1 − α k · · · η n η − 1 k α n − α k η − 1 k ℓ P s =1 f k ( k − ℓ + s ) a s 1 · · · η − 1 k ℓ P s =1 f k ( k − ℓ + s ) a sℓ        = V f G 1 = V [ E k | B ] . It’s easy t o kno w that F and V are b oth nonsingular o v er F q , and so F G 1 and f G 1 are b oth the generator matrices of the co de GRL k ( α , 1 , A ℓ × ℓ ) . Note that A ℓ × ℓ ∈ GL ℓ ( F q ), it means that a 11 , a 21 , · · · , a ℓ 1 are not all equal to zero, and then without loss of generalit y , w e can supp ose that a ℓ 1 6 = 0. Th us, if f G 1 generates a 24 RS co de, then b y L emma 2 .4 , for the ( i, n − k + 1)-th en try of B (1 ≤ i ≤ k ), there exist α n +1 , · · · , α n + ℓ ∈ F q \ { α 1 , . . . , α n } such that η − 1 i ℓ X s =1 f i ( k − ℓ + s ) a s 1 = η n +1 η − 1 i α n +1 − α i , where η n +1 = k Q s =1 ( α n +1 − α s ) , i.e., η n +1 α n +1 − α i = ℓ X s =1 f i ( k − ℓ + s ) a s 1 = ℓ X s =1 ℓ − s X t =0 ( − 1) t σ t α ℓ − s − t i ! a s 1 . It’s means that α 1 , . . . , α k ( k > ℓ ) are distinct r o ots of the p olynomial η n +1 = ( α n +1 − x ) ℓ X s =1 ℓ − s X t =0 ( − 1) t σ t x ℓ − s − t ! a s 1 , whic h is contradict with deg  ( α n +1 − x ) ℓ P s =1  ℓ − s P t =0 ( − 1) t σ t x ℓ − s − t  a s 1  = ℓ . Therefore f G 1 is not a generator ma t r ix for an y RS co de, i.e., the co de GRL k ( α , 1 , A ℓ × ℓ )( k > ℓ ) is non-RS. F urthermore, the co de G RL k ( α , v , A ℓ × ℓ )( k > ℓ ) is non-GRS. This completes the pro of o f Theorem 5 . 1 .  Remark 5.1 (1) If k = ℓ and A ℓ × ℓ is an ℓ × ℓ V andermon d e matrix       1 · · · 1 β 1 · · · β ℓ β 2 1 · · · β 2 ℓ . . . . . . . . . β ℓ − 1 1 · · · β ℓ − 1 ℓ       , then the c orr esp o nding c o de GRL k ( α , v , A ℓ × ℓ ) has the fol lowing gener ate matrix       1 · · · 1 1 · · · 1 α 1 · · · α n β 1 · · · β ℓ α 2 1 · · · α 2 n β 2 1 · · · β 2 ℓ . . . . . . . . . . . . . . . . . . α ℓ − 1 1 · · · α ℓ − 1 n β ℓ − 1 1 · · · β ℓ − 1 ℓ       , (5.2) it’s e a s y to know that the c o de G RL k ( α , v , A ℓ × ℓ ) gen e r ate d by ( 5 . 2 ) is a n [ n + ℓ, ℓ, n + 1] RS c o de w i th the evaluation-p o int se quenc e α = ( α 1 , . . . , α n , β 1 , . . . , β ℓ ) . (2) If k = ℓ and A ℓ × ℓ is a ℓ × ℓ non-si n gular lower triangular matrix     a 11 0 · · · 0 ∗ a 22 · · · 0 . . . . . . . . . . . . ∗ ∗ · · · a ℓℓ     with a ii ∈ F ∗ q , then the c orr esp onding c o de G R L k ( α , v , A ℓ × ℓ ) has the fol lowing gen er ate m atrix     1 · · · 1 a 11 0 · · · 0 α 1 · · · α n ∗ a 22 · · · 0 . . . . . . . . . . . . . . . . . . . . . α ℓ − 1 1 · · · α ℓ − 1 n ∗ ∗ · · · a ℓℓ     , (5.3) 25 By the Magma pr o gr am, for the c o de G RL k ( α , v , A ℓ × ℓ ) ge ner ate d by ( 5 . 3 ) , we have dim  GRL 2 k ( α , v , A ℓ × ℓ )  > 2 k − 1 , then by Pr op osition 1 of the r efe r enc e [ 24 ], the c o de GRL k ( α , v , A ℓ × ℓ ) ge ner ate d by ( 5 . 3 ) is a non-RS typ e. 6 Conclus ions In this pap er, w e study t he L CD prop erty of the co de G R L k ( α , 1 , A ℓ × ℓ ) with b oth Eu- clidean and Hermitian inner pro ducts. By taking some sp ecial v ector α = ( α 1 , . . . , α n ), w e construct sev eral classes o f Euclidean LCD GRL co des, Hermitian LCD GRL co des, G RL co des with small-dimensional hull, resp ectiv ely . And f or sev eral classes of Hermitian GRL co des, w e firstly giv e an upp er b ound for the dimension o f the h ull. Subseq uen tly , we apply the ab ov e results to construct sev eral classes of EAQEC Cs. Fina lly , w e prov e that the co de GRL k ( α , 1 , A ℓ × ℓ ) is non-GRS for k > ℓ . Ac knowledgemen t This pa p er is supp or ted b y National Natural Science F oundation of China (Gran t No. 12471494 ) and Natural Science F o undation o f Sic h uan Pro vince (2024NSFSC2051) . References [1] Ala hmadi A, Altassan A, AlKenani A, et al. A multise cret-sharing sc heme based on LCD co des[J]. Mathematics, 2020, 8 ( 2): 272 . [2] Ab dukhalik ov K, Ding C, V erma G K. Some constructions of non-generalized Reed- Solomon MDS Co des[J]. arXiv preprint arXiv:2506.04080, 2025. 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App endix App endix A : Examples for Section 3 In this section, we g ive some examples for Theorems 3 .1 , 3.4 , 3.6 , 3 .7 and 3.9 , where Examples A.1 - A.2 are for Theorems 3.1 (1)- (2), Examples A.3 - A.4 are for Theorem 3.4 ( 1 )- (2); Example A.5 is for Theorem 3.6 ; Example A.6 (1)-(3) a nd Examples A.7 - A.8 a r e fo r Theorem 3.7 (2),(4), (5), (1) and (3); Example A.9 is fo r Theorem 3.9 , resp ective ly . Example A.1 L et q = 3 4 , k = 5 , δ = 2 , F ∗ q = h γ i , A 2 × 2 =  γ γ 2 γ 3 γ 5  . By α i = γ q − 1 k i = γ 16 i , we have α =  γ 16+2 , γ 32+2 , γ 48+2 , γ 64+2 , γ 80+2  , and then the c orr esp o n ding GRL c o de C has the fol lowing ge ner ator matrix       1 1 1 1 1 0 0 γ 18 γ 34 γ 50 γ 66 γ 82 0 0 ( γ 18 ) 2 ( γ 34 ) 2 ( γ 50 ) 2 ( γ 66 ) 2 ( γ 82 ) 2 0 0 ( γ 18 ) 3 ( γ 34 ) 3 ( γ 50 ) 3 ( γ 66 ) 3 ( γ 82 ) 3 γ γ 2 ( γ 18 ) 4 ( γ 34 ) 4 ( γ 50 ) 4 ( γ 66 ) 4 ( γ 82 ) 4 γ 3 γ 5       Base d on the Magm a pr o gr am, C is Euclide an LCD MD S with the p ar ameters [7 , 5 , 3] 3 4 , wh ich is c onsistent with The or em 3.1 (1). Example A.2 L et q = 5 2 , k = 8 , δ = 1 , F ∗ q = h γ i , A 4 × 4 =    1 γ γ 2 γ γ γ 3 γ 5 γ 7 γ γ 6 γ 10 γ 14 γ 3 γ 9 γ 15 γ 21    . By α i = γ q − 1 k i = γ 3 i , we ha v e α =  γ 3+1 , γ 6+1 , γ 9+1 , γ 12+1 , γ 15+1 , γ 18+1 , γ 21+1 , γ 24+1  , and then the c orr esp o n ding GRL c o de C has the fol lowing ge ner ator matrix             1 1 1 1 1 1 1 1 0 0 0 0 γ 4 γ 7 γ 10 γ 13 γ 16 γ 19 γ 22 γ 25 0 0 0 0 ( γ 4 ) 2 ( γ 7 ) 2 ( γ 10 ) 2 ( γ 13 ) 2 ( γ 16 ) 2 ( γ 19 ) 2 ( γ 22 ) 2 ( γ 25 ) 2 0 0 0 0 ( γ 4 ) 3 ( γ 7 ) 3 ( γ 10 ) 3 ( γ 13 ) 3 ( γ 16 ) 3 ( γ 19 ) 3 ( γ 22 ) 3 ( γ 25 ) 3 0 0 0 0 ( γ 4 ) 4 ( γ 7 ) 4 ( γ 10 ) 4 ( γ 13 ) 4 ( γ 16 ) 4 ( γ 19 ) 4 ( γ 22 ) 4 ( γ 25 ) 4 1 γ γ 2 γ ( γ 4 ) 5 ( γ 7 ) 5 ( γ 10 ) 5 ( γ 13 ) 5 ( γ 16 ) 5 ( γ 19 ) 5 ( γ 22 ) 5 ( γ 25 ) 5 γ γ 3 γ 5 γ 7 ( γ 4 ) 6 ( γ 7 ) 6 ( γ 10 ) 2 ( γ 13 ) 6 ( γ 16 ) 6 ( γ 19 ) 6 ( γ 22 ) 6 ( γ 25 ) 6 γ γ 6 γ 10 γ 14 ( γ 4 ) 7 ( γ 7 ) 7 ( γ 10 ) 7 ( γ 13 ) 7 ( γ 16 ) 7 ( γ 19 ) 7 ( γ 22 ) 7 ( γ 25 ) 7 γ 3 γ 9 γ 15 γ 21             29 Base d on the Magma pr o gr a m, C is Euclide an LCD NMDS w i th the p ar a m eters [12 , 8 , 4] 5 2 , which is c on sistent with The or em 3.1 (2). Example A.3 L et q = 5 2 , k = 12 , δ = 2 , F ∗ q = h γ i , A 2 × 2 =   γ 2 γ 4 γ 6 γ 2 γ 5 γ 7 γ 5 γ 8 γ 9   . By α i = γ q − 1 k i = γ 2 i , we ha v e α =  0 , γ 2+2 , γ 4+2 , γ 6+2 , γ 8+2 , γ 10+2 , γ 12+2 , γ 14+2 , γ 16+2 , γ 18+2 , γ 20+2 , γ 22+2 , γ 24+2  , and then the c orr esp o n ding GRL c o de C has the fol lowing ge ner ator matrix                   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 γ 4 γ 6 γ 8 γ 10 γ 12 γ 14 γ 16 γ 18 γ 20 γ 22 γ 24 γ 26 0 0 0 0 γ 8 γ 12 γ 16 γ 20 γ 24 γ 28 γ 32 γ 36 γ 40 γ 44 γ 48 γ 52 0 0 0 0 γ 12 γ 18 γ 24 γ 30 γ 36 γ 42 γ 48 γ 54 γ 60 γ 66 γ 72 γ 78 0 0 0 0 γ 16 γ 24 γ 32 γ 40 γ 48 γ 56 γ 64 γ 72 γ 80 γ 88 γ 96 γ 104 0 0 0 0 γ 20 γ 30 γ 40 γ 50 γ 60 γ 70 γ 80 γ 90 γ 100 γ 110 γ 120 γ 130 0 0 0 0 γ 24 γ 36 γ 48 γ 60 γ 72 γ 84 γ 96 γ 108 γ 120 γ 132 γ 144 γ 156 0 0 0 0 γ 28 γ 42 γ 56 γ 70 γ 84 γ 98 γ 112 γ 126 γ 140 γ 154 γ 168 γ 182 0 0 0 0 γ 32 γ 48 γ 64 γ 80 γ 96 γ 112 γ 128 γ 144 γ 160 γ 176 γ 192 γ 208 0 0 0 0 γ 36 γ 54 γ 72 γ 90 γ 108 γ 126 γ 144 γ 162 γ 180 γ 198 γ 216 γ 234 γ 2 γ 4 γ 6 0 γ 40 γ 60 γ 80 γ 100 γ 120 γ 140 γ 160 γ 180 γ 200 γ 220 γ 240 γ 260 γ 2 γ 5 γ 7 0 γ 44 γ 66 γ 88 γ 110 γ 132 γ 154 γ 176 γ 198 γ 220 γ 242 γ 264 γ 286 γ 5 γ 8 γ 9                   Base d on the Magma pr o gr am , C is Euclide an LCD NMDS w ith the p ar ame ters [1 6 , 12 , 4] 5 2 , which is c on sistent with The or em 3.4 (1). Example A.4 L et q = 5 2 , k = 8 , δ = 1 , F ∗ q = h γ i , A 2 × 2 =  1 γ γ 2 γ 4  . By α i = γ q − 1 k i = γ 3 i , we have α =  0 , γ 3+1 , γ 6+1 , γ 9+1 , γ 12+1 , γ 15+1 , γ 18+1 , γ 21+1 , γ 24+1  , and then the c orr esp o n ding GRL c o de C has the fol lowing ge ner ator matrix            1 1 1 1 1 1 1 1 1 1 1 1 1 0 γ 4 γ 7 γ 10 γ 13 γ 16 γ 19 γ 22 γ 25 0 0 0 0 0 γ 8 γ 14 γ 20 γ 26 γ 32 γ 38 γ 44 γ 50 0 0 0 0 0 γ 12 γ 21 γ 30 γ 39 γ 48 γ 57 γ 66 γ 75 0 0 0 0 0 γ 16 γ 28 γ 40 γ 52 γ 64 γ 76 γ 88 γ 100 γ 1 γ 3 γ 5 γ 7 0 γ 20 γ 35 γ 50 γ 65 γ 80 γ 95 γ 110 γ 125 γ 3 γ 4 γ 8 γ 9 0 γ 24 γ 42 γ 60 γ 78 γ 96 γ 114 γ 132 γ 150 γ 11 γ 12 γ 13 γ 15 0 γ 28 γ 49 γ 70 γ 91 γ 112 γ 133 γ 154 γ 175 γ 16 γ 20 γ 21 γ 23            Base d on the Magma pr o gr a m, C is Euclide an LCD NMDS w i th the p ar a m eters [13 , 8 , 5] 5 2 , which is c on sistent with The or em 3.4 (2). 30 Example A.5 L et q = 31 , k = 5 , s = 1 , t = 9 , F ∗ q = h γ i , A 2 × 2 =  γ γ 2 γ 3 γ 5  , it’s e asy to se e that q − 1 k = 6 ∤ t − s = 8 v 2 ( q − 1) − v 2 ( k ) − 1 = 0 6 = v 2 ( t − s ) = 3 . Then we have α =  γ 6+1 , γ 12+1 , γ 18+1 , γ 24+1 , γ 30+1 , γ 6+9 , γ 12+9 , γ 18+9 , γ 24+9 , γ 30+9  , and then the c orr esp o n ding GRL c o de C has the fol lowing ge ner ator matrix       1 1 1 1 1 1 1 1 1 1 0 0 γ 7 γ 13 γ 19 γ 25 γ 31 γ 15 γ 21 γ 27 γ 33 γ 39 0 0 ( γ 7 ) 2 ( γ 13 ) 2 ( γ 19 ) 2 ( γ 25 ) 2 ( γ 31 ) 2 ( γ 15 ) 2 ( γ 21 ) 2 ( γ 27 ) 2 ( γ 33 ) 2 ( γ 39 ) 2 0 0 ( γ 7 ) 3 ( γ 13 ) 3 ( γ 19 ) 3 ( γ 25 ) 3 ( γ 31 ) 3 ( γ 15 ) 3 ( γ 21 ) 3 ( γ 27 ) 3 ( γ 33 ) 3 ( γ 39 ) 3 γ γ 2 ( γ 7 ) 4 ( γ 13 ) 4 ( γ 19 ) 4 ( γ 25 ) 4 ( γ 31 ) 4 ( γ 15 ) 4 ( γ 21 ) 4 ( γ 27 ) 4 ( γ 33 ) 4 ( γ 39 ) 4 γ 3 γ 5       . Base d on the Magma pr o gr a m, C is Euclide an LCD NMDS with the p ar ameters [12 , 5 , 7] 31 , which is c on sistent with The or em 3.6 . Example A.6 L et q = 3 4 , k = 4 , F ∗ q = h γ i , A 2 × 2 =  γ γ 2 γ 3 γ 5  , it’s e asy to se e that v 2 ( k ) = 2 < v 2 ( q − 1) = 4 and v 5 ( k ) = 0 < v 5 ( q − 1 ) = 1 , then w e have the fol low ing thr e e choic es (1) L et δ = 2 v 2 ( q − 1) − v 2 ( k ) − 2 = 1 . By α i = γ q − 1 k i = γ 20 i , we have α =  γ 20 , γ 40 , γ 60 , γ 80 , γ 21 , γ 41 , γ 61 , γ 81  , and then the c orr esp o n ding GRL c o de C has the fol lowing gen er ator m a trix    1 1 1 1 1 1 1 1 0 0 γ 20 γ 40 γ 60 γ 80 γ 21 γ 41 γ 61 γ 81 0 0 γ 40 γ 80 γ 120 γ 160 γ 42 γ 82 γ 122 γ 162 γ γ 2 γ 60 γ 120 γ 180 γ 240 γ 63 γ 123 γ 183 γ 243 γ 3 γ 5    Base d on the Magma pr o gr am, C is Euclide an LCD NMDS with the p ar ameters [10 , 4 , 6] 3 4 , which is c onsistent with The or e m 3.7 (2). (2) L et δ = 2 v 2 ( q − 1)+1 = 2 5 . By α i = γ q − 1 k i = γ 20 i , we ha v e α =  γ 20 , γ 40 , γ 60 , γ 80 , γ 52 , γ 72 , γ 92 , γ 112  , 31 and then the c orr esp o n ding GRL c o de C has the fol lowing gen er ator m a trix    1 1 1 1 1 1 1 1 0 0 γ 20 γ 40 γ 60 γ 80 γ 52 γ 72 γ 92 γ 112 0 0 γ 40 γ 80 γ 120 γ 160 γ 104 γ 144 γ 184 γ 224 γ γ 2 γ 60 γ 120 γ 180 γ 240 γ 156 γ 216 γ 276 γ 336 γ 3 γ 5    Base d on the Mag m a pr o gr am, C is Euclide an LCD MDS with the p ar am eters [10 , 4 , 7] 3 4 , which is c onsistent with The or e m 3.7 (4). (3) L et δ = 5 v 2 ( q − 1) − v 2 ( k ) − 1 = 1 . By α i = γ q − 1 k i = γ 20 i , we have α =  γ 20 , γ 40 , γ 60 , γ 80 , γ 21 , γ 41 , γ 61 , γ 81  , and then the c orr esp o n ding GRL c o de C has the fol lowing gen er ator m a trix    1 1 1 1 1 1 1 1 0 0 γ 20 γ 40 γ 60 γ 80 γ 21 γ 41 γ 61 γ 81 0 0 γ 40 γ 80 γ 120 γ 160 γ 42 γ 82 γ 122 γ 162 γ γ 2 γ 60 γ 120 γ 180 γ 240 γ 63 γ 123 γ 183 γ 243 γ 3 γ 5    Base d on the Magma pr o gr am, C is Euclide an LCD NMDS with the p ar ameters [10 , 4 , 6] 3 4 , which is c onsistent with The or e m 3.7 (5). Example A.7 L et q = 5 2 , k = 8 , F ∗ q = h γ i , A 2 × 2 =  1 γ γ 2 γ 4  , it’s e asy to se e that v 2 ( k ) = v 2 ( q − 1 ) = 3 . Then by taking δ = 2 1 and α i = γ q − 1 k i = γ 3 i , we have α =  γ 3 , γ 6 , γ 9 , γ 12 , γ 15 , γ 18 , γ 21 , γ 24 , γ 4 , γ 7 , γ 10 , γ 13 , γ 16 , γ 19 , γ 22 , γ 25  , and then the c orr esp o n ding GRL c o de C has the fol lowing ge ner ator matrix            1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 γ 3 γ 6 γ 9 γ 12 γ 15 γ 18 γ 21 γ 24 γ 4 γ 7 γ 10 γ 13 γ 16 γ 19 γ 22 γ 25 0 0 γ 6 γ 12 γ 18 γ 24 γ 30 γ 36 γ 42 γ 48 γ 8 γ 14 γ 20 γ 26 γ 32 γ 38 γ 44 γ 50 0 0 γ 9 γ 18 γ 27 γ 36 γ 45 γ 54 γ 63 γ 72 γ 12 γ 21 γ 30 γ 39 γ 48 γ 57 γ 66 γ 75 0 0 γ 12 γ 24 γ 36 γ 48 γ 60 γ 72 γ 84 γ 96 γ 16 γ 28 γ 40 γ 52 γ 64 γ 76 γ 88 γ 100 0 0 γ 15 γ 30 γ 45 γ 60 γ 75 γ 90 γ 105 γ 120 γ 20 γ 35 γ 50 γ 65 γ 80 γ 95 γ 110 γ 125 0 0 γ 18 γ 36 γ 54 γ 72 γ 90 γ 108 γ 126 γ 144 γ 24 γ 42 γ 60 γ 78 γ 96 γ 114 γ 132 γ 150 1 γ γ 21 γ 42 γ 63 γ 84 γ 105 γ 126 γ 147 γ 168 γ 28 γ 49 γ 70 γ 91 γ 112 γ 133 γ 154 γ 175 γ 2 γ 4            Base d on the Magma pr o gr am , C is Euclide an LCD AMDS w ith the p ar ame ters [1 8 , 8 , 10] 5 2 , which is c on sistent with The or em 3.7 (1). Example A.8 L et q = 7 2 , k = 6 , F ∗ q = h γ i , A 2 × 2 =  γ γ 2 γ 3 γ 5  , it’s e asy to se e that v 2 ( q − 1) − v 2 ( k ) = 2 6 = 1 32 and v 3 ( k ) = v 3 ( q − 1 ) = 1 . Then by taking δ = 3 1 and α i = γ q − 1 k i = γ 2 i , we have α =  γ 2 , γ 4 , γ 6 , γ 8 , γ 10 , γ 12 , γ 5 , γ 7 , γ 9 , γ 11 , γ 13 , γ 15  , and then the c orr esp o n ding GRL c o de C has the fol lowing ge ner ator matrix        1 1 1 1 1 1 1 1 1 1 1 1 0 0 γ 2 γ 4 γ 6 γ 8 γ 10 γ 12 γ 5 γ 7 γ 9 γ 11 γ 13 γ 15 0 0 γ 4 γ 8 γ 12 γ 16 γ 20 γ 24 γ 10 γ 14 γ 18 γ 22 γ 26 γ 30 0 0 γ 6 γ 12 γ 18 γ 24 γ 30 γ 36 γ 15 γ 21 γ 27 γ 33 γ 39 γ 45 0 0 γ 8 γ 16 γ 24 γ 32 γ 40 γ 48 γ 20 γ 28 γ 36 γ 44 γ 52 γ 60 γ γ 2 γ 10 γ 20 γ 30 γ 40 γ 50 γ 60 γ 25 γ 35 γ 45 γ 55 γ 65 γ 75 γ 3 γ 5        . Base d on the Magma pr o gr a m, C is Euclide an LCD NMDS w i th the p ar a m eters [14 , 6 , 8] 7 2 , which is c on sistent with The or em 3.7 (3). Example A.9 L et q = 7 2 , k = 12 , F ∗ q = h γ i , A 2 × 2 =  γ γ 2 γ 3 γ 5  , it’s e asy to se e that (3 k , q ) = 1 , q − 1 = 48 / ∈ { k , 2 k , 3 k } = { 12 , 2 4 , 3 6 } , and v 2 ( q − 1) = v 2 ( k ) = 4 ≥ 1 . Then by taking α i = γ q − 1 k i = γ 4 i , we ha v e α =  γ 4 , γ 8 , . . . , γ 44 , γ 48 , γ 5 , γ 9 , . . . , γ 45 , γ 49 , γ 6 , γ 10 , . . . , γ 46 , γ 50  , Base d on the Magma pr o gr am, C is Euclide an LCD with the p ar ameters [50 , 12] 7 2 , which is c onsistent with The o r em 3.9 . App endix B : Examples for Section 4 In this App endix B, we giv e some examples for Theorems 4.1 , 4.4 , 4.7 , 4.8 a nd 4.12 , where Example A.10 (1)-( 2 ) ar e for The orem 4.1 (1)- (2); Ex amples A.11 - A.12 are fo r Theorem 4.4 (1)-(2); Examples A.13 - A.15 is f o r Theorems 4.7 , 4.8 and 4.12 , r espective ly . Example A.10 L et q = 3 2 , k = 8 , δ = 1 , F ∗ q 2 = h γ i . By α i = γ q 2 − 1 k i = γ 10 i , we ha v e α =  γ 11 , γ 21 , γ 31 , γ 41 , γ 51 , γ 61 , γ 71  , and then we have the fol lowing two c ases. 33 (1) By taking A 2 × 2 =  γ γ 2 γ 3 γ 5  . The n the c orr es p onding GRL c o de C has the fol lowing gener ator matrix            1 1 1 1 1 1 1 1 0 0 γ 11 γ 21 γ 31 γ 41 γ 51 γ 61 γ 71 γ 81 0 0 ( γ 11 ) 2 ( γ 21 ) 2 ( γ 31 ) 2 ( γ 41 ) 2 ( γ 51 ) 2 ( γ 61 ) 2 ( γ 71 ) 2 ( γ 81 ) 2 0 0 ( γ 11 ) 3 ( γ 21 ) 3 ( γ 31 ) 3 ( γ 41 ) 3 ( γ 51 ) 3 ( γ 61 ) 3 ( γ 71 ) 3 ( γ 81 ) 3 0 0 ( γ 11 ) 4 ( γ 21 ) 4 ( γ 31 ) 4 ( γ 41 ) 4 ( γ 51 ) 4 ( γ 61 ) 4 ( γ 71 ) 4 ( γ 81 ) 4 0 0 ( γ 11 ) 5 ( γ 21 ) 5 ( γ 31 ) 5 ( γ 41 ) 5 ( γ 51 ) 5 ( γ 61 ) 5 ( γ 71 ) 5 ( γ 81 ) 5 0 0 ( γ 11 ) 6 ( γ 21 ) 6 ( γ 31 ) 6 ( γ 41 ) 6 ( γ 51 ) 6 ( γ 61 ) 6 ( γ 71 ) 6 ( γ 81 ) 6 γ γ 2 ( γ 11 ) 7 ( γ 21 ) 7 ( γ 31 ) 7 ( γ 41 ) 7 ( γ 51 ) 7 ( γ 61 ) 7 ( γ 71 ) 7 ( γ 81 ) 7 γ 3 γ 5            Base d on the Magma pr o gr am , C is Hermitian LCD MDS with the p ar a m eters [10 , 8 , 3] 3 4 , which is c onsistent with The or em 4.1 (1) . (2) By taking A 4 × 4 =    γ 1 1 1 1 γ 2 1 1 1 1 γ 3 1 γ 4 γ 5 γ 6 γ 7    . Then the c orr esp onding GRL c o de C has the fol lowing gener ator matrix            1 1 1 1 1 1 1 1 0 0 0 0 γ 11 γ 21 γ 31 γ 41 γ 51 γ 61 γ 71 γ 81 0 0 0 0 ( γ 11 ) 2 ( γ 21 ) 2 ( γ 31 ) 2 ( γ 41 ) 2 ( γ 51 ) 2 ( γ 61 ) 2 ( γ 71 ) 2 ( γ 81 ) 2 0 0 0 0 ( γ 11 ) 3 ( γ 21 ) 3 ( γ 31 ) 3 ( γ 41 ) 3 ( γ 51 ) 3 ( γ 61 ) 3 ( γ 71 ) 3 ( γ 81 ) 3 0 0 0 0 ( γ 11 ) 4 ( γ 21 ) 4 ( γ 31 ) 4 ( γ 41 ) 4 ( γ 51 ) 4 ( γ 61 ) 4 ( γ 71 ) 4 ( γ 81 ) 4 γ 1 1 1 ( γ 11 ) 5 ( γ 21 ) 5 ( γ 31 ) 5 ( γ 41 ) 5 ( γ 51 ) 5 ( γ 61 ) 5 ( γ 71 ) 5 ( γ 81 ) 5 1 γ 2 1 1 ( γ 11 ) 6 ( γ 21 ) 6 ( γ 31 ) 6 ( γ 41 ) 6 ( γ 51 ) 6 ( γ 61 ) 6 ( γ 71 ) 6 ( γ 81 ) 6 1 1 γ 3 1 ( γ 11 ) 7 ( γ 21 ) 7 ( γ 31 ) 7 ( γ 41 ) 7 ( γ 51 ) 7 ( γ 61 ) 7 ( γ 71 ) 7 ( γ 81 ) 7 γ 4 γ 5 γ 6 γ 7            Base d on the Magma pr o gr am, C is Hermitian LCD NMDS with the p ar am eters [1 2 , 8 , 4] 3 4 , which is c on sistent with The or em 4.1 (2) . Example A.11 L et q = 3 3 , k = 13 , δ = 0 , F ∗ q 2 = h γ i . By α i = γ q 2 − 1 k i = γ 56 i , we have α =  0 , γ 56 , ( γ 56 ) 2 , ( γ 56 ) 3 , ( γ 56 ) 4 , ( γ 56 ) 5 , ( γ 56 ) 6 , ( γ 56 ) 7 , ( γ 56 ) 8 , ( γ 56 ) 9 , ( γ 56 ) 10 , ( γ 56 ) 11 , ( γ 56 ) 12 , ( γ 56 ) 13  . By taking A 2 × 2 =  γ γ 2 γ 3 γ 5  . Th en the c orr esp ondin g GRL c o de C has the f o l lowing gene r ator matrix            1 1 1 . . . 1 0 0 0 γ 56 ( γ 56 ) 2 . . . ( γ 56 ) 13 0 0 0 ( γ 56 ) 2 (( γ 56 ) 2 ) 2 . . . (( γ 56 ) 13 ) 2 0 0 0 ( γ 56 ) 3 (( γ 56 ) 2 ) 3 . . . (( γ 56 ) 13 ) 3 0 0 . . . . . . . . . . . . . . . . . . . . . 0 ( γ 56 ) 10 (( γ 56 ) 2 ) 10 . . . (( γ 56 ) 13 ) 10 0 0 0 ( γ 56 ) 11 (( γ 56 ) 2 ) 11 . . . (( γ 56 ) 13 ) 11 γ 1 γ 2 0 ( γ 56 ) 12 (( γ 56 ) 2 ) 12 . . . (( γ 56 ) 13 ) 12 γ 3 γ 5            13 × 16 Base d on the Magma pr o gr am , C is Hermi tian LCD MDS with the p ar ameters [16 , 13 , 4] 3 4 , which is c on sistent with The or em 4.4 (1) . 34 Example A.12 L et q = 5 2 , k = 8 , δ = 0 , F ∗ q 2 = h γ i . By α i = γ q 2 − 1 k i = γ 78 i , we ha v e α =  0 , γ 78 , ( γ 78 ) 2 , ( γ 78 ) 3 , ( γ 78 ) 4 , ( γ 78 ) 5 , ( γ 78 ) 6 , ( γ 78 ) 7 , ( γ 78 ) 8  . By taking A 2 × 2 =  γ γ 2 γ 3 γ 5  . Th en the c orr esp ondin g GRL c o de C has the f o l lowing gene r ator matrix           1 1 1 1 1 1 1 1 1 0 0 0 0 0 γ 78 ( γ 78 ) 2 ( γ 78 ) 3 ( γ 78 ) 4 ( γ 78 ) 5 ( γ 78 ) 6 ( γ 78 ) 7 ( γ 78 ) 8 0 0 0 0 0 ( γ 78 ) 2 (( γ 78 ) 2 ) 2 (( γ 78 ) 3 ) 2 (( γ 78 ) 4 ) 2 (( γ 78 ) 5 ) 2 (( γ 78 ) 6 ) 2 (( γ 78 ) 7 ) 2 (( γ 78 ) 8 ) 2 0 0 0 0 0 ( γ 78 ) 3 (( γ 78 ) 2 ) 3 (( γ 78 ) 3 ) 3 (( γ 78 ) 4 ) 3 (( γ 78 ) 5 ) 3 (( γ 78 ) 6 ) 3 (( γ 78 ) 7 ) 3 (( γ 78 ) 8 ) 3 0 0 0 0 0 ( γ 78 ) 4 (( γ 78 ) 2 ) 4 (( γ 78 ) 3 ) 4 (( γ 78 ) 4 ) 4 (( γ 78 ) 5 ) 4 (( γ 78 ) 6 ) 4 (( γ 78 ) 7 ) 4 (( γ 78 ) 8 ) 4 γ 1 1 1 0 ( γ 78 ) 5 (( γ 78 ) 2 ) 5 (( γ 78 ) 3 ) 5 (( γ 78 ) 4 ) 5 (( γ 78 ) 5 ) 5 (( γ 78 ) 6 ) 5 (( γ 78 ) 7 ) 5 (( γ 78 ) 8 ) 5 1 γ 2 1 1 0 ( γ 78 ) 6 (( γ 78 ) 2 ) 6 (( γ 78 ) 3 ) 6 (( γ 78 ) 4 ) 6 (( γ 78 ) 5 ) 6 (( γ 78 ) 6 ) 6 (( γ 78 ) 7 ) 6 (( γ 78 ) 8 ) 6 1 1 γ 3 1 0 ( γ 78 ) 7 (( γ 78 ) 2 ) 7 (( γ 78 ) 3 ) 7 (( γ 78 ) 4 ) 7 (( γ 78 ) 5 ) 7 (( γ 78 ) 6 ) 7 (( γ 78 ) 7 ) 7 (( γ 78 ) 8 ) 7 γ 4 γ 5 γ 6 γ 7           . Base d on the Magma pr o gr am, C is Hermitian LCD NMDS with the p ar am eters [1 3 , 8 , 5] 5 4 , which is c on sistent with The or em 4.4 (2) . Example A.13 L et q = 13 , k = 6 , F ∗ q 2 = h γ i , A ℓ × ℓ ∈ L =     γ γ 2 γ 3 γ 5  ,   γ 1 γ 3 γ 5 γ 3 γ 4 γ 8 γ 11 γ 12 γ 13      . It’s e asy to se e that q 2 − 1 k = 28 and 0 / ∈ T =  v 2 ( q 2 − 1 ) − 1 − v 2 ( i + ( k − i ) q ) | 1 ≤ i ≤ k − 1  = { 1 } , then w e take some ( s , t ) satisfy q 2 − 1 k ∤ s − t a n d v 2 ( s − t ) / ∈ T , for ex a mple, ( s, t ) ∈ K = { ( 4 , 1) , (5 , 2) , (10 , 1) , (11 , 2) , ( 2 8 , 1) } , furthermor e, we have α ∈  γ 28+ s , γ 56+ s , · · · , γ 140+ s , γ 168+ s , γ 28+ t , γ 56+ t , · · · , γ 140+ t , γ 168+ t  | ( s, t ) ∈ K  . Then the c orr es p onding GRL c o de C has the fol low ing gener a tor matrix         1 1 · · · 1 1 1 1 · · · 1 1 γ 28+ s γ 56+ s · · · γ 140+ s γ 168+ s γ 28+ t γ 56+ t · · · γ 140+ t γ 168+ t ( γ 28+ s ) 2 ( γ 56+ s ) 2 · · · ( γ 140+ s ) 2 ( γ 168+ s ) 2 ( γ 28+ t ) 2 ( γ 56+ t ) 2 · · · ( γ 140+ t ) 2 ( γ 168+ t ) 2 ( γ 28+ s ) 3 ( γ 56+ s ) 3 · · · ( γ 140+ s ) 3 ( γ 168+ s ) 3 ( γ 28+ t ) 3 ( γ 56+ t ) 3 · · · ( γ 140+ t ) 3 ( γ 168+ t ) 3 ( γ 28+ s ) 4 ( γ 56+ s ) 4 · · · ( γ 140+ s ) 4 ( γ 168+ s ) 4 ( γ 28+ t ) 4 ( γ 56+ t ) 4 · · · ( γ 140+ t ) 4 ( γ 168+ t ) 4 ( γ 28+ s ) 5 ( γ 56+ s ) 5 · · · ( γ 140+ s ) 5 ( γ 168+ s ) 5 ( γ 28+ t ) 5 ( γ 56+ t ) 5 · · · ( γ 140+ t ) 5 ( γ 168+ t ) 5 0 (6 − ℓ ) × ℓ A ℓ × ℓ         with A ℓ × ℓ ∈ L. Base d on the Magma pr o gr am , w hen A ℓ × ℓ =  γ γ 2 γ 3 γ 5  , C is Hermi tian LCD NMDS with the p a r ameters [14 , 6 , 8 ] 13 2 , which is c onsistent w ith The or e m 4.7 ; w hen 35 A ℓ × ℓ =   γ 1 γ 3 γ 5 γ 3 γ 4 γ 8 γ 11 γ 12 γ 13   , C is Hermitian LCD NMDS with the p ar ameters [15 , 6 , 9] 13 2 , exp e ct for the c ase ( s, t ) = (5 , 2) , which is Hermitian LCD with the p ar ameters [15 , 6 , 8] 13 2 , which is c onsistent with The o r em 4.7 . Example A.14 L et q = 11 , k = 5 , F ∗ q 2 = h γ i , A 2 × 2 =  γ γ 2 γ 3 γ 5  . It’s e as y to se e that q 2 − 1 k = 24 and T =  v 2 ( q 2 − 1) − 1 − v 2 ( i + ( k − i ) q ) | 1 ≤ i ≤ k − 1  = { 2 } , then we take some δ satisfy q 2 − 1 k ∤ δ and v 2 ( δ ) / ∈ T , for example, δ = 2 , 3 , 6 , 8 , furthermor e , we have α ∈  γ 24 , γ 48 , γ 72 , γ 96 , γ 120 , γ 24+ δ , γ 48+ δ , γ 72+ δ , γ 96+ δ , γ 120+ δ  | δ = 2 , 3 , 6 , 8  . Then the c orr es p onding GRL c o de C has the fol low ing gener a tor matrix       1 1 1 1 1 1 1 1 1 1 0 0 γ 24 γ 48 γ 72 γ 96 γ 120 γ 24+ δ γ 48+ δ γ 72+ δ γ 96+ δ γ 120+ δ 0 0 ( γ 24 ) 2 ( γ 48 ) 2 ( γ 72 ) 2 ( γ 96 ) 2 ( γ 120 ) 2  γ 24+ δ  2  γ 48+ δ  2  γ 72+ δ  2  γ 96+ δ  2  γ 120+ δ  2 0 0 ( γ 24 ) 3 ( γ 48 ) 3 ( γ 72 ) 3 ( γ 96 ) 3 ( γ 120 ) 3  γ 24+ δ  3  γ 48+ δ  3  γ 72+ δ  3  γ 96+ δ  3  γ 120+ δ  3 γ γ 2 ( γ 24 ) 4 ( γ 48 ) 4 ( γ 72 ) 4 ( γ 96 ) 4 ( γ 120 ) 4  γ 24+ δ  4  γ 48+ δ  4  γ 72+ δ  4  γ 96+ δ  4  γ 120+ δ  4 γ 3 γ 5       Base d on the Magma pr o gr am, C is Hermitian LCD NMDS with the p ar ameters [1 2 , 5 , 7] 11 2 , which is c on sistent with The or em 4.8 . Example A.15 L et q = 11 , k = 5 , F ∗ q 2 = h γ i , A 2 × 2 =  γ γ 2 γ 3 γ 5  . I t’s e as y to kno w that  q 2 − 1 ( q 2 − 1 , i + ( k − i ) q )     1 ≤ i ≤ k − 1  = { 8 , 24 } , then we take some δ satisfy ( δ k , q ) = 1 and q 2 − 1 ( q 2 − 1 ,i +( k − i ) q ) ∤ δ + 1 for any 1 ≤ i ≤ k − 1 , fo r example, δ = 1 , 2 , 3 , 4 , 5 , 6 , furthermor e, we have α ∈  γ 24 , γ 48 , γ 72 , γ 96 , γ 120 , γ 25 , γ 49 , γ 73 , γ 97 , γ 121 , . . . , γ 24+ δ , γ 48+ δ , γ 72+ δ , γ 96+ δ , γ 120+ δ  | δ = 1 , 2 , 3 , 4 , 5 , 6  . Then the c orr es p onding GRL c o de C has the fol low ing gener a tor matrix      1 · · · 1 1 · · · 1 · · · 1 · · · 1 0 0 γ 24 · · · γ 120 γ 24+1 · · · γ 120+1 · · · γ 24+ δ · · · γ 120+ δ 0 0  γ 24  2 · · ·  γ 120  2  γ 24+1  2 · · ·  γ 120+1  2 · · ·  γ 24+ δ  2 · · ·  γ 120+ δ  2 0 0  γ 24  3 · · ·  γ 120  3  γ 24+1  3 · · ·  γ 120+1  3 · · ·  γ 24+ δ  3 · · ·  γ 120+ δ  3 γ γ 2  γ 24  4 · · ·  γ 120  4  γ 24+1  4 · · ·  γ 120+1  4 · · ·  γ 24+ δ  4 · · ·  γ 120+ δ  4 γ 3 γ 5      Base d on the Magma pr o gr am , C is Hermitian LCD NMDS with the p ar ameters { [12 , 5 , 7 ] 11 2 , [17 , 5 , 12] 11 2 , [22 , 5 , 17] 11 2 , [27 , 5 , 22] 11 2 , [32 , 5 , 27] 11 2 , [37 , 5 , 32] 11 2 } , which is c on sistent with The or em 4.12 . 36