On Expanded Cyclic Codes

On Expanded Cyclic Co des Yingquan W u Link A Media Devices Corp. 2550 W alsh Av e, Suite 200 San ta Clara, CA 95051, USA Octob er 24, 2021 Abstract The pap er has a threefold purpos e . The first purp ose is to pr esent an explicit descr iption o f ex- panded cyclic co des defined in GF( q m ). The prop o s ed explicit constr uction of expanded gener ator matrix and expanded pa rity chec k ma trix maintains the sym b ol-wise alg ebraic structure and thus keeps many impo rtant or iginal characteristics . The second purp ose of this pap er is to identify a class of cons tant-w eight cy clic co des. Specifically , we sho w that a well-known class of q -ary BCH co des e xcluding the all-z e ro co deword are constant-w eight cyclic co des. Moreov er, we show this class o f co des a chiev e the Plotkin b ound. The las t purp ose of the pa p er is to character iz e expanded cyclic codes utilizing the pr op osed expanded generator matrix and parity chec k matr ix. W e c har- acterize the pr op erties of comp onent co dewords of a co deword and particularly identify the precise conditions under which a c o deword can b e repr esented by a subbasis . Our developmen ts r eveal a n alternative while mor e gener al view on the subspace sub co des of Reed- Solomon co des. With the new insights, we prese nt an improved low e r b o und on the minimu m dista nce of an expanded cyclic co de by exploiting the ge neralized co ncatenated structure. W e a lso show that the fixed-ra te binar y expanded Reed-Solo mon co des ar e asymptotically “bad” , in the sense that the ra tio o f minimu m distance over co de leng th diminishes with co de le ngth going to infinit y . It ov erturns the prev alent conjecture that they ar e “go o d” codes and deviates from the ensemble of genera lized Reed-Solo mo n co des which as ymptotically achiev es the Gilbert-V arshamov b ound. 1 I. In tro duction The pap er has a threefold purp ose, the first of whic h is to presen t an explicit description of expand ed cyclic co des defined in GF( q m ). An explicit construction of an expan d ed generator matrix is giv en in terms of trace and dual/complemen tary basis [13, 14]. An expan d ed parity c hec k matrix can b e constructed th rough r ep lacing eac h element with its matrix repr esen tation [14]. Th e prop osed explicit construction of expand ed generator matrix and exp anded parity c hec k matrix main tains the symb ol- wise algebraic structure and th us k eeps many imp ortan t original prop erties. The second purp ose of this pap er is to iden tify a class of constan t-w eigh t cyclic co des. Constan t- w eigh t co des ha v e b een used in a num b er of applications, includ ing co d e-division multiple- access (CDMA) systems for optical fib ers, automatic-rep eat-request error-con trol s ystems, parallel asyn - c hronous comm unications, etc. A et al established [1] a general theorem to obtain a binary constan t- w eigh t cyclic co de from a p -ary linear cyclic co de, where p is a prime, b y using a representati on of GF( p ) as cyclic sh ifts of a binary p -tup le, and constru ctions were deriv ed for four classes of b inary constan t-w eigh t co des. Bitan and Etzion [4] constructed optimal constan t w eigh t cyclically p erm utable co des w ith wei ght w and minim um Hamming distance 2 w − 2. Xing an d Ling [21] constructed a class of constant-w eigh t co des b y emp lo ying the narr o w ray class groups of alge braic curve s. Chee and Ling [7] in tro duced a new com binatorial construction f or q -ary constan t-w eigh t co des which yields sev eral families of optimal co d es and asym p totically optimal co des. The encyclop edic w ork on the lo wer b ounds, through explicit constructions, of constant-w eigh t co des was presented in [6] whereas the collectiv e u p p er b oun ds for constan t-w eigh t co des wa s in v estigated in [2]. In this pap er, w e sho w that a well-kno wn class of q -ary BCH co d es excluding th e all-zero co dewo rd are constan t-w eigh t cyclic co des. Moreo v er, w e sho w this class of co des ac hiev e the Plotkin b oun d (cf. [2, 3]). The th ird and final pur p ose of the pap er is to c haracterize the p rop osed expanded generator matrix and parity c hec k matrix. In literature, researc h has m ainly fo cused on th e bin ary realization of Reed- Solomon co des, which has b een app lied in v arious practices, e.g., in magnetic r ecordin g and optical data storage. Retter sho w ed [15] that the ensem ble of generalized Reed-Solomon co d es ac hiev e th e Gilb ert-V arsharmo v b ound , whic h represents the b est-known asymptotic lo w er b ound of th e ratio of m inim um distance d to co de length n th at b in ary co des of any rate exist (cf. [3]). In [16], the orthogonalit y of binary expans ions of Reed-Solomon cod es is c h aracterized in terms of their s p ectra and the b ases used to exp and them. In [17], it is sh o wn that the binary we igh t en umerator of a Reed- Solomon co des ov er GF(2 m ) as well as the gaps of weigh t distribu tion generally dep end on the c h oice of basis. Th e binary we igh t enumeratio n of particular r ealizat ions of sp ecial Reed-Solomon co des has b een studied in [5, 11, 12]. V ard y and Be’ery [20] sh o w ed that high-rate Reed-Solomon co d es con tain BCH sub co d es, and subsequently exploited this prop erty to reduce the trellis complexit y of bit-lev el soft- decision maxim um-lik eliho o d (Viterbi) deco ding. S eguin [19] c haracterized the conditions under which 2 an expanded cyclic co de is also cyclic. The author extended the su b co d e concept dev elop ed in [20] to more general concept of p rimary comp onent (where a su b co d e is treated as a trivial comp onen t). Based on the generalized concatenate d stru cture presented in [19], S ak adibara and Kasahara derived a low er b oun d on the minimum distance of expanded cyclic co des. Hattori et al [10] c haracterized the dimension of su bspace sub co des of Reed-Solomon co des un der c haracteristic 2. C ui and Pe i [8] en tended the Hattori’s wo rk to general GF( q m ) and to generalized Reed-Solomon co des. In this pap er, we sho w the pr ecise conditions u nder w h ic h a co d ew ord can b e presente d by a subb asis. Ou r dev elopmen ts immed iately reve als an alternativ e formula of the dimension of sub space sub co d es of the Reed-Solomon co des defined in [10]. Moreo v er, the d ev elopmen ts hand ily answe r some of op en problems listed in the end of [10], includ ing th e determination of b est su bspace and extension to general fi eld GF( q m ). With th e new insights, we present a lo wer b ound on the minimum distance of an exp anded cyclic co de, exploiting the generalized concatenated stru cture w hic h can b e viewed as an impro v ement o ve r the b ound given in [18]. In [18], the minim um distance of an outer co d e is sh o wn to b e b ound ed by the largest n umb er of consecutiv e conjugacy elements, whereas our dev elopmen ts pro vide true m inim um distance of th e outer co de, which effectiv ely tak es into accoun t for the b asis realizatio n. W e also sho w th at the b inary image of Reed-Solomon co d es is asymptotically “bad”, in the sense that the ratio of minim um distance o ver co de length diminishes with co d e length going to infinity . It o v erturn s th e w ell-kno wn conjecture that they are “go o d” cod es (cf. [9]) and d eviates from the ensem ble of generalized Reed-Solomon cod es which asymptotically ac hiev es the Gilb ert-V arshamo v b ound [15]. I I. Description of Expanded Cyclic Co des Denote b y GF( q m ) a Galo is field, where q is a p o w er of a prime, and GF ∗ ( q m ) △ = GF( q m ) \{ 0 } . Let α denote a primitiv e elemen t in GF( q m ). Let G ( x ) = ( x − α 1 )( x − α 2 )( x − α 3 ) ... ( x − α R ) (1) b e the generator p olynomial of the primitiv e cycli c code C ( N , K ), where N = q m − 1 an d R = N − K . It is w ell-kno wn that the p arity c heck matrix can b e represent ed in the form of (cf. [3]) H ( α 1 , α 2 , . . . , α R ) =           1 α 1 1 α 2 1 α 3 1 . . . α N − 2 1 α N − 1 1 1 α 1 2 α 2 2 α 3 2 . . . α N − 2 2 α N − 1 2 1 α 1 3 α 2 3 α 3 3 . . . α N − 2 3 α N − 1 3 . . . . . . . . . . . . . . . . . . . . . 1 α 1 R α 2 R α 3 R . . . α N − 2 R α N − 1 R           . (2) Denote g ( γ ) △ = [1 , γ , γ 2 , . . . , γ N − 1 ] , (3) 3 and its corresp ond ing p olynomial g γ ( x ) △ = 1 + γ x + γ 2 x 2 + . . . + γ N − 1 x N − 1 . (4) It can b e easily sho wn that g γ ( x ) = ( γ x ) N − 1 γ x − 1 = ( γ x − 1)( γ x − α )( γ x − α 2 ) . . . ( γ x − α N − 1 ) γ x − 1 = ( γ x − α )( γ x − α 2 ) . . . ( γ x − α N − 1 ) = γ ( N − 1) ( x − γ − 1 α 1 )( x − γ − 1 α 2 ) . . . ( x − γ − 1 α N − 1 ) = γ − 1 ( x − γ − 1 α 1 )( x − γ − 1 α 2 ) . . . ( x − γ − 1 α N − 1 ) (5) where the second “=” is due to x N − 1 = ( x − α 0 )( x − α 1 )( x − α 2 ) . . . ( x − α N − 2 )( x − α N − 1 ) . Lemma 1 L et { γ i : 1 ≤ i ≤ K } △ = GF ∗ ( q m ) \{ α − 1 j : 1 ≤ j ≤ R } (6) Then, the C ( N , K ) c o de define d by (1) has the fol lowing gener ator matrix G ( γ 1 , γ 2 , . . . , γ K ) =           1 γ 1 1 γ 2 1 γ 3 1 . . . γ N − 2 1 γ N − 1 1 1 γ 1 2 γ 2 2 γ 3 2 . . . γ N − 2 2 γ N − 1 2 1 γ 1 3 γ 2 3 γ 3 3 . . . γ N − 2 3 γ N − 1 3 . . . . . . . . . . . . . . . . . . . . . 1 γ 1 K γ 2 K γ 3 K . . . γ N − 2 K γ N − 1 K           . (7) Pr o of: Evidentl y , the ab o ve matrix exhibits full rank d u e to th e V ander m onde prop er ty . On the other hand, as indicated b y (5 ), th e p olynomials asso ciated with eac h ro w of G conta in ro ots α 1 , α 2 , α 3 , . . . , α R , subsequentl y divide G ( x ). ✷✷ It is w orth clarifying that a generator p olynomial is directly asso ciated w ith a parit y c hec k matrix, whereas a parit y c hec k p olynomial is directly asso ciated with a generator matrix. Corollary 1 The R e e d-Solomon c o de define d by the ge ner ator p olynomial G ( x ) = ( x − α δ )( x − α δ +1 )( x − α δ +2 ) . . . ( x − α δ + R − 1 ) (8) has gener ator matrix G ( α − δ +1 , α − δ +2 , . . . , α − δ + K ) , as define d in (7) . 4 Let { β 1 , β 2 , . . . , β m } b e a basis of GF( q m ). An elemen t γ ∈ GF( q m ) can b e decomp osed in f orm of γ = µ 1 β 1 + µ 2 β 2 + . . . + µ m β m , (9) where µ i ∈ GF( q ). The follo win g theorem pr esen ts an explicit construction of generator m atrix and parit y c hec k matrix of an exp anded co d e, which main tains the symb ol-wise algebraic structure an d thus k eeps man y imp ortant original prop er ties. Theorem 1 ( i ) . L et C ( N , K ) b e define d in GF ( q m ) and with g e ner ator matrix G = [ g 1 , g 2 , . . . , g K ] T . Then, its exp ansion c o de in GF ( q ) under a b asis { β i } m i =1 has the fol lowing g ener ator matrix G e = [ β 1 g 1 , . . . , β m g 1 , β 1 g 2 , . . . , β m g 2 , . . . . . . , β 1 g K , . . . , β m g K ] T . ( ii ) . L et C ( N , K ) b e define d in GF ( q m ) and with p arity che ck matrix H T = [ h 1 , h 2 , h 3 , . . . , h N ] T . Then, its exp ansion c o de in GF ( q ) under a b asis { β i } m i =1 has the fol lowing p arity che ck matrix H T e = [ β 1 h 1 , . . . , β m h 1 , β 1 h 2 , . . . , β m h 2 , . . . . . . , β 1 h N , . . . , β m h N ] T . Note that the subscript “ e ” is used to denote the corresp onding expansion. Pr o of: ( i ). W e first sho w the matrix G e is full rank through con tradiction. Let m X j =1 g 1 ν 1 ,j β j + m X j =1 g 2 ν 2 ,j β j + . . . + m X j =1 g K ν K,j β j = 0 , where ν i,j ∈ GF( q ) and at least one ν i,j is nont rivial, and g i β j is viewed as an mN -dim en sion v ector o v er the base field GF( q ). Let θ i = m X j =1 ν i,j β j , i = 1 , 2 , . . . , K. Then, w e ha v e θ 1 g 1 + θ 2 g 2 + . . . + θ K g K = 0 , whic h is con tradictory to Lemma 1 w here r o ws in (7 ) are linearly ind ep endent (herein θ i g ( γ i ) is view ed as an N -dimension v ector o v er th e extension field GF( q m )). On the other hand, a cod ew ord can b e repr esen ted b y the linear com bination of rows in (7), say , c = θ ′ 1 g 1 + θ ′ 2 g 2 + . . . + θ ′ K g K . 5 Note θ ′ i can b e represen ted by the basis β j , j = 1 , 2 , . . . , m, sa y , θ ′ i = m X j =1 ν ′ i,j β j , i = 1 , 2 , . . . , K. Then, the expansion of the co deword c can b e decomp osed as ro ws of exp an d ed generator matrix defined in (10) c = m X j =1 g 1 ν ′ 1 ,j β j + m X j =1 g 2 ν ′ 2 ,j β j + . . . + m X j =1 g K ν ′ K,j β j . The pro of of ( ii ) follo w s the observ ation b elo w [ h 1 , h 2 , h 3 , . . . , h N ] · [ c 1 , c 2 , c 3 , . . . , c N ] T = N X i =1 h i c i = N X i =1 h i m X j =1 µ j ( c i ) β j = N X i =1 m X j =1 µ j ( c i ) h i β j = [ h 1 β 1 , . . . , h 1 β m , . . . , h N β 1 , . . . , h N β m ] · [ µ 1 ( c 1 ) , . . . , µ m ( c 1 ) , . . . , µ 1 ( c N ) , . . . , µ m ( c N )] T , where µ i ( γ ) ∈ GF( q ) denotes the co efficient asso ciated with β i in th e d ecomp osition of γ ∈ GF( q m ). ✷✷ Corollary 2 L et β 1 , β 2 , . . . , β m b e a b asis of GF ( q m ) . ( i ) . The gener ator matrix of the exp ansion of the cyclic c o de define d by (7) is G e ( γ 1 , γ 2 , . . . , γ K ) =                             β 1 γ 1 1 β 1 γ 2 1 β 1 γ 3 1 β 1 . . . γ N − 2 1 β 1 γ N − 1 1 β 1 . . . . . . . . . . . . . . . . . . . . . β m γ 1 1 β m γ 2 1 β m γ 3 1 β m . . . γ N − 2 1 β m γ N − 1 1 β m β 1 γ 1 2 β 1 γ 2 2 β 1 γ 3 2 β 1 . . . γ N − 2 2 β 1 γ N − 1 2 β 1 . . . . . . . . . . . . . . . . . . . . . β m γ 1 2 β m γ 2 2 β m γ 3 2 β m . . . γ N − 2 2 β m γ N − 1 2 β m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . β 1 γ 1 K β 1 γ 2 K β 1 γ 3 K β 1 . . . γ N − 2 K β 1 γ N − 1 K β 1 . . . . . . . . . . . . . . . . . . . . . β m γ 1 K β m γ 2 K β m γ 3 K β m . . . γ N − 2 K β m γ N − 1 K β m                             (10) 6 ( ii ) . The p arity che ck matrix of the exp ansion of the cyclic c o de define d by (1) is H T e ( α 1 , α 2 , . . . , α R ) =                             β 1 β 1 β 1 . . . β 1 β 1 . . . . . . . . . . . . . . . . . . β m β m β m . . . β m β m β 1 α 1 β 1 α 2 β 1 α 3 . . . β 1 α R − 1 β 1 α R . . . . . . . . . . . . . . . . . . β m α 1 β m α 2 β m α 3 . . . β m α R − 1 β m α R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . β 1 α N − 1 1 β 1 α N − 1 2 β 1 α N − 1 3 . . . β 1 α N − 1 R − 1 β 1 α N − 1 R . . . . . . . . . . . . . . . . . . β m α N − 1 1 β m α N − 1 2 β m α N − 1 3 . . . β m α N − 1 R − 1 β m α N − 1 R                             (11) In Section IV we will characte rize the expanded cod es based on the pr op osed expanded generato r matrix (whereas the analysis straigh tforwa rdly applies to the prop osed expanded parit y c hec k matrix). I I I. A Class of Constan t-W eigh t Cyclic Co d es The follo win g lemma identifies a su b field elemen t (cf. [13]). Lemma 2 An element γ i n GF ( q m ) lies in the sub fie ld GF ( q ) i f and only if γ q = γ . F or instance, γ = α 33 ∈ GF(2 10 ) lies in the subfi eld GF(2 5 ), as γ 32 = γ . Let p γ ( x ) denote the minimal p olynomial of γ ∈ GF ( q m ), wh ich is defined as th e minimum-degree nominal p olynomial which h as all co efficient s p er taining to GF( q ) and cont ains the root γ . Let m γ b e the minimal dimension of γ , whic h is defined as the minim um n umber satisfying γ q m γ = γ (note that γ can b e represente d by an m γ -dimensional v ector in GF( q )). It is worth n oting that m γ is a factor of m . It is well- known that (cf. [13]) the minimal p olynomial of γ ∈ GF( q m ) ov er GF ( q ) can b e explicitly expressed by p γ ( x ) = ( x − γ )( x − γ q )( x − γ q 2 ) . . . ( x − γ q m γ − 1 ) . (12) where m γ denotes the minimal dimension of γ . Moreo v er, the conjugacy class, φ ( γ ) △ = { γ , γ q , γ q 2 , . . . , γ q m γ − 1 } (13) share the minimal p olynomial p γ ( x ) (cf. [13]). 7 Denote b y w Hamming weigh t and w ( c ) th e Hamming w eigh t of a ve ctor c . Moreo v er, denote by w γ ( c ) the wei ght of c con tributed b y γ , i.e., w γ ( c ) △ = |{ i : c i = γ }| . The follo win g theorem iden tifies a class of constan t-w eigh t co d es. Theorem 2 L et γ b e a non-subfield e lement in GF ( q m ) and C ( N , m ) b e asso ciate d with the gener ator p olynomial G ( x ) = x N − 1 p γ ( x ) , wher e p γ ( x ) i s the minimal p olynomia l of γ over GF ( q ) as define d in (12) . Then, C ∗ △ = C \{ 0 } is a c o de of c onstant weight q m − 1 ( q − 1) . Mor e over, e ach element of GF ∗ ( q ) app e ars exactly q m − 1 times in e ach c o dewor d. Pr o of: W e observe that th e generator p olynomial G ( x ) con tains consecutiv e ro ots, γ q m − 1 +1 , γ q m − 1 +2 , . . . , γ N − 1 , γ N = 1. Thus, its min im um distance is at least q m − q m − 1 . On the other h and, note that G ′ ( x ) = x N − 1 ( x − 1) p γ ( x ) con tains the consecutive ro ots, γ q m − 1 +1 , γ q m − 1 +2 , . . . , γ N − 1 . T h us, the co d e C ′ ( N , m + 1) asso ciated with the generator p olynomial G ′ ( x ) has minimum distance at least q m − q m − 1 − 1. Let c b e a co dew ord in C ∗ . Assume th at th e nonzero elemen t ν ∗ ∈ GF ∗ ( q ) cont ribu tes the most w eigh t to c , i.e., w ν ∗ ( c ) ≥ w ν ( c ) , ∀ ν ∈ GF ∗ ( q ) . Since its Hamming w eigh t is at least q m − q m − 1 , w e ha v e w ν ∗ ( c ) ≥ q m − q m − 1 q − 1 = q m − 1 . W e observe that c ′ = c − ν ∗ · 1 is a v alid co d eword in C ′ , wh ere 1 d enotes the all-one co deword. Note c ′ flips all zero elemen ts of c to − ν ∗ while all ν ∗ elemen ts of c to zero. Ther efore, we obtain q m − q m − 1 − 1 ≤ w ( c ′ ) = w ( c ) + w 0 ( c ) − w ν ∗ ( c ) = q m − 1 − w ν ∗ ( c ) whic h immed iate manifests w ν ∗ ( c ) ≤ q m − 1 . Consequ en tly , it holds w ν ∗ ( c ) = q m − 1 . Finally , th e prop erty w ( c ) ≥ q m − 1 ( q − 1) holds if and only if w ν ( c ) = q m − 1 , ∀ ν ∈ GF ∗ ( q ) , and subsequently , w ( c ) = q m − 1 ( q − 1) , ∀ c , where eac h of q − 1 element s in GF ∗ ( q ) equally con tributes weigh t q m − 1 . The th eorem is concluded. ✷✷ 8 Let α b e a p rimitiv e element in GF(2 4 ). F ollo wing Th eorem 2, the non zero co d ew ords asso ciated with the generator p olynomial x 15 − 1 p α − 1 ( x ) ha v e constant weigh t 2 4 − 1 = 8, as listed b elo w. [ 1 1 1 1 0 1 0 1 1 0 0 1 0 0 0 ] [ 0 1 1 1 1 0 1 0 1 1 0 0 1 0 0 ] [ 1 0 0 0 1 1 1 1 0 1 0 1 1 0 0 ] [ 0 0 1 1 1 1 0 1 0 1 1 0 0 1 0 ] [ 1 1 0 0 1 0 0 0 1 1 1 1 0 1 0 ] [ 0 1 0 0 0 1 1 1 1 0 1 0 1 1 0 ] [ 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0 ] [ 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1 ] [ 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 ] [ 1 0 0 1 0 0 0 1 1 1 1 0 1 0 1 ] [ 0 1 1 0 0 1 0 0 0 1 1 1 1 0 1 ] [ 1 1 0 1 0 1 1 0 0 1 0 0 0 1 1 ] [ 0 0 1 0 0 0 1 1 1 1 0 1 0 1 1 ] [ 1 0 1 0 1 1 0 0 1 0 0 0 1 1 1 ] [ 0 1 0 1 1 0 0 1 0 0 0 1 1 1 1 ] Theorem 3 L et γ ∈ GF ( q m ) b e a subfield element. L et C ( N , m γ ) b e define d by the gene r ator p olyno- mial x N − 1 p γ ( x ) , wher e p γ ( x ) i s the minimal p olynomial as define d in (12 ) . Then, C ∗ △ = C \{ 0 } is a c o de of c onstant weight q m γ − 1 ( q − 1) q m − 1 q m γ − 1 , wher e m γ denotes the minimal dimension of γ . Mor e over, e ach element of GF ∗ ( q ) app e ars exactly q m γ − 1 q m − 1 q m γ − 1 times in e ach c o dewor d. Pr o of: Lemma 1 in conjun ction with Corollary 2 indicates that the expanded generator matrix is, G e = [ β 1 g ( γ − 1 ) , . . . , β m g ( γ − 1 ) , β 1 g ( γ − q ) , . . . , β m g ( γ − q ) , . . . , β 1 g ( γ − q m γ − 1 ) , . . . , β m g ( γ − q m γ − 1 )] T . W e obser ve that eac h ro w is p erio dic with duration q m γ − 1, and thus contai ns q m − 1 q m γ − 1 p erio d s (note that m γ = m γ − 1 ). This shows that eac h exp an d ed co dewo rd is also p erio dic and conta ins q m − 1 q m γ − 1 p erio d s. It can b e easily seen that γ is a primitive elemen t in the su bfield GF( q m γ ) and eac h p erio d of an expanded co de is exactly a co dewo rd asso ciated w ith the generator p olynomial x ( q m γ − 1) − 1 p γ ( x ) defined in the subfi eld GF( q m γ ). T h us, eac h p erio d of a co de has constant weig ht q m γ − 1 ( q − 1), follo wing Theorem 2. ✷✷ Let α b e a p rimitiv e element in GF(2 6 ). F ollo wing Th eorem 3, the non zero co d ew ords asso ciated with the generator p olynomial x 63 − 1 p α − 9 ( x ) ha v e constant weigh t 2 3 − 1 · 2 6 − 1 2 3 − 1 = 36, as listed b elo w. [ 1011 100 1011100 1011100 1011100 1011100 1011100 1011100 1011100 10111 00 ] [ 1110 010 1110010 1110010 1110010 1110010 1110010 1110010 1110010 11100 10 ] [ 0101 110 0101110 0101110 0101110 0101110 0101110 0101110 0101110 01011 10 ] [ 0111 001 0111001 0111001 0111001 0111001 0111001 0111001 0111001 01110 01 ] [ 1100 101 1100101 1100101 1100101 1100101 1100101 1100101 1100101 11001 01 ] [ 1001 011 1001011 1001011 1001011 1001011 1001011 1001011 1001011 10010 11 ] [ 0010 111 0010111 0010111 0010111 0010111 0010111 0010111 0010111 00101 11 ] The Plotkin b ound asserts that the minim um distance d min of an y (linear or n onlinear) co de w hic h has A co dew ords of length N o v er th e alphab et of size q is b ou n ded by (cf. [3]) d min ≤ N ( q − 1) q − q A − 1 . 9 Theorem 4 ( i ) . The c yc lic c o de C ( N , m γ ) asso ci ate d with gener ator p olynomial x N − 1 p γ ( x ) exactly matches the P lotkin b ound, wher e m γ denotes the minimal dimension of γ . ( ii ) . Given that γ is a non-subfield element in GF ( q m ) , the cyclic c o de C ( N , m + 1) asso ciate d with gener ator p olynomial x N − 1 ( x − 1) p γ ( x ) matches the Plotkin b ound. ( iii ) . Given that γ is a non-subfie ld element in GF ( q m ) , the c onstant-weight cyclic c o de C ∗ ( N , m ) asso ciate d with the gener ator p olynomial x N − 1 p γ ( x ) matches the Plotkin b ound. Pr o of: ( i ). When γ is a non-sub field elemen t, w e hav e N ( q − 1) q − q A − 1 = ( q m − 1)( q − 1) q − q ( q m ) − 1 = q m − 1 ( q − 1) = d min , where the cod e size A = q m . When γ is a subfield elemen t, we again h a v e N ( q − 1) q − q A − 1 = ( q m − 1)( q − 1) q − q ( q m γ ) − 1 = q m γ − 1 ( q − 1) q m − 1 q m γ − 1 = d min , where the cod e size A = q m γ . ( ii ). Note that the minimum distance is precisely q m − q m − 1 − 1. Consequent ly ,  N ( q − 1) q − q A − 1  =  ( q m − 1)( q − 1) q − q ( q m +1 ) − 1  = q m − 1 ( q − 1) − 1 +  2 − q − 1 − q − m q − q − m  = q m − 1 ( q − 1) − 1 = d min , where the cod e size A = q m +1 . The pro of of part ( ii ) follo w s  N ( q − 1) q − q A − 1  =  ( q m − 1)( q − 1) q − q ( q m − 1) − 1  =  q m − 1 ( q − 1) + q − 1 q ( q m − 2)  = q m − 1 ( q − 1) = d min , where the cod e size A = q m − 1. ✷✷ IV. Characterization of Exp anded Cyclic Co des In this sectio n, we carry out analysis on expan d ed generator matrix G e . I t is straigh tforwa rd to sho w that all results hold in analogue to expanded parit y c hec k matrix H e . Giv en a basis { β i } m i =1 in GF( q m ), denote by µ i ( γ ) ∈ GF( q ) the function of γ that represent s its expansion asso ciated with basis β i , i.e., the v alue µ i in (9). F or br evit y , the fun ction µ i ( · ) is also applied to a v ector y = [ y 1 , y 2 , . . . , y n ] ∈ GF( q m ) n , suc h that µ i ( y ) △ = [ µ i ( y 1 ) , µ i ( y 2 ) , µ i ( y 3 ) , . . . , µ i ( y n )] . (14) and to a p olynomial f ( x ) = f 0 + f 1 x + f 2 x 2 + . . . + f n x n ∈ GF( q m )[ x ], such that µ i ( f ( x )) △ = µ i ( f 0 ) + µ i ( f 1 ) · x + µ i ( f 2 ) · x 2 + . . . + µ i ( f n ) · x n . (15) 10 Let β i 1 β i 2 ...β i k = m X j =1 µ ( i 1 ,i 2 ,...,i k ) j · β j , (16) where µ ( i 1 ,i 2 ,...,i k ) j , j = 1 , 2 , . . . , m , are regarded as constan ts since { β i } m i =1 are treated as kno wn pa- rameters, and for brevit y β q s i = m X j =1 µ ( i [ q s ]) j β j . (17) Lemma 3 Give n a non-subfield element γ in GF ∗ ( q m ) , the elements of g ( γ ) c annot b e gener ate d by a su bb asis. Note that subbasis is a w eak er concept than s ubfield. The b asis of a su bfield can b e expand ed to repr esen t the whole field, thus a su bfield corresp onds a sub basis, whereas the a su bbasis is not necessarily asso ciated with a subfi eld. Pr o of: W e pro v e it b y con tradiction. Let { β 1 , β 2 , . . . , β m } b e the basis of GF( q m ) and { β i 1 , β i 2 , . . . , β i k } ( k < m ) b e the su bbasis for the element s of g ( γ ). C onsequen tly , the linear span (un der addition op- eration) of the multiplic ativ e group { 1, γ , γ 2 , . . . , γ N − 1 } is a field, with basis { β i 1 , β i 2 , . . . , β i k } . It means that th e span (under addition) of { β i 1 , β i 2 , . . . , β i k } , wh ich has q k elemen ts, is a p rop er subfield of GF( q m ). It follo ws that γ is sub field elemen t, violating the assumption. ✷✷ According to (5), β i g γ ( x ) can b e divided in to β i g γ ( x ) = x N − 1 p γ − 1 ( x ) β i χ γ ( x ) , where x N − 1 p γ − 1 ( x ) ∈ GF ( q )[ x ], and χ γ ( x ) = γ − 1 ( x − γ − q )( x − γ − q 2 ) . . . ( x − γ − q − 1+ m γ ) (18) where m γ = m γ − 1 (recall that it is defi ned as the smallest num b er su c h th at γ q m γ = γ ). Decomp ose β i g γ ( x ) in to β i g γ ( x ) = m X j =1 β j · µ j ( β i χ γ ( x )) x N − 1 p γ − 1 ( x ) . The ab o v e expression immediately manifests that µ j ( β i g γ ( x )) = µ j ( β i χ γ ( x )) x N − 1 p γ − 1 ( x ) , (19) The follo win g theorem summarizes the significant prop erty of G e ( γ ). Theorem 5 ( i ) . Give n γ a non-subfield element in GF ∗ ( q m ) , no nontrivial c o dewor d asso ciate d with the gener ator p olynomial G e ( γ ) c an b e r epr esente d b y a subb asis. M or e over, the c omp onent wor ds 11 µ j ( β i g ( γ )) , 1 ≤ i, j ≤ m , ar e c o dewor ds asso ciate d with the gener ator p olynomial x N − 1 p γ − 1 ( x ) and exhibit c onstant Hamming weight q m − 1 ( q − 1) . ( ii ) . Given γ a subfield element with minimal dimension m γ < m , the c omp onent wor ds µ j ( β i g ( γ )) , 1 ≤ i, j ≤ m , ar e c o dewor ds asso ciate d with the g e ner ator p olynomial x N − 1 p γ − 1 ( x ) , and exhibit Hamming weight q m γ − 1 ( q − 1) q m − 1 q m γ − 1 or zer o. M or e over, a c omp onent c o dewor d µ j ( β i g ( γ )) is the al l-ze r o wor d if and only i f j 6 = i and β i GF ( q m γ ) c an b e r epr esente d b y the subb asis { β 1 , . . . , β j − 1 , β j +1 , . . . , β m } . Pr o of: W e shall only sho w the part related to subbasis representa tion. ( i ). A co d ew ord c can b e exp ressed as c = θ g ( γ ) for some θ ∈ GF ∗ ( q m ). Assume that θ g ( γ ) is generated by a subbasis { β i 1 , β i 2 , . . . , β i k } out of { β i } m i =1 ( k < m ), then, g ( γ ) is generated by the subbasis { θ − 1 β i 1 , θ − 1 β i 2 , . . . , θ − 1 β i k } (out of the alternativ e basis { θ − 1 β i } m i =1 ). This clearly conflicts to Lemma 3, whic h asserts that g ( γ ) cannot b e generated by a sub b asis. ( ii ). Clearly , a straigh tforw ard equiv alence is that a comp onen t co dewo rd µ j ( β i g ( γ )) is the all-zero w ord if and on ly if β i g ( γ ) can b e r epresen ted by the su bbasis { β 1 , . . . , β j − 1 , β j +1 , . . . , β m } . Th e condi- tion j 6 = i is due to the f act that β j can not b e represented by the su bbasis { β 1 , . . . , β j − 1 , β j +1 , . . . , β m } . On the other hand, Lemma 3 ind icates that the field GF ( q m γ ) is the closure of g ( γ ) under the addition op eration, therefore, if β i g ( γ ) is repr esented by a su b basis, then β i GF( q m γ ) is also r epresen ted b y the subbasis. ✷✷ Note that the constant-w eigh t c haracterization follo w s Th eorem 2 an d m γ = m γ − 1 . W e give t w o examples to illustrate part ( ii ). Let the comp osite b asis { β 1 , β 2 , β 3 , β 4 } = { 1, α 5 , α , α 6 } b e employ ed to expand g ( α 10 ) in GF(2 4 ). W e ha v e µ i ( β j g ( α 10 )) = 0 , i = 3 , 4 , j = 1 , 2 , or i = 1 , 2 , j = 3 , 4 . Alternativ ely , let the basis b e { β 1 , β 2 , β 3 , β 4 } = { 1 + α , α 5 , α , α 6 } , wh ere the subfi eld GF(2 2 ) is represent ed by th e subbasis { 1 + α , α 5 , α } . Consequently , w e ha v e, µ i ( β j g ( α 10 )) = 0 , i = 4 , j = 2 , or i = 1 , 2 , j = 3 , 4 . Note that the ab o v e theorem ju stifies that the density of binary expanded p arit y chec k matrix of a Reed-Solomon co d e is near on e half, due to d ominan t n on-subfield elemen ts w hose corresp ond ing densit y is precisely 2 ( m − 1) 2 m − 1 . Corollary 3 If a p olynomial p ( x ) ∈ GF ( q )[ x ] divides the gene r ator p olynomial G ( x ) ∈ GF ( q m )[ x ] , then it also divides the al l c omp onent wor d p olynomials µ i ( c ( x )) , i = 1 , 2 , . . . , m , wher e c ( x ) denotes a c o dewor d ge ne r ate d by G ( x ) . Mor e over, let d b e the minimum distanc e of the c o de asso ciate d with the g ener ator p olynomial p ( x ) , then the weight of a nonzer o c omp onent c o dewor d, µ i ( c ) , is at le ast d . 12 Pr o of: Let G ( x ) = p ( x ) G ′ ( x ) and c ( x ) = a ( x ) G ( x ). In analogue to (19), we h av e µ i ( c ( x )) = µ i ( a ( x ) G ( x )) = µ i ( a ( x ) G ′ ( x ) p ( x )) = µ i ( a ( x ) G ′ ( x )) · p ( x ) for i = 1 , 2 , . . . , m . ✷✷ W e observe that γ β i = m X j =1 µ j ( γ ) β j β i , 1 ≤ i ≤ m. (20) Substituting (16) into the ab o v e expression, we obtain γ β i = m X l =1 β l m X j =1 µ ( i,j ) l µ j ( γ ) = m X l =1 β l · f i,l ( µ 1 ( γ ) , µ 2 ( γ ) , . . . , µ m ( γ )) (21) where the coefficient function f i,l ( µ 1 , µ 2 , . . . , µ m ) △ = m X j =1 µ ( i,j ) l µ j , i = 1 , 2 , . . . , m. (22) The follo win g lemma c haracterizes the prop erties of linear function f i,j ( µ 1 , µ 2 , . . . , µ m ). Lemma 4 ( i ) . F or any given j , ther e do esnot exist nontrivial { ν i } m i =1 ∈ GF ( q ) such that m X i =1 ν i f i,j ( µ 1 ( γ ) , µ 2 ( γ ) , . . . , µ m ( γ )) = 0 , ∀ γ ∈ GF ∗ ( q m ) . ( ii ) . Given γ a non-subfield element in GF ∗ ( q m ) , the m ve ctors [ f i,j ( µ 1 (1) , . . . , µ m (1)) , f i,j ( µ 1 ( γ ) , . . . , µ m ( γ )) , . . . , f i,j ( µ 1 ( γ N − 1 ) , . . . , µ m ( γ N − 1 ))  , i = 1 , 2 , . . . , m ar e line arly i ndep endent. Pr o of: ( i ). Su pp ose it is not true, sa y , m X i =1 ν i f i,j ( µ 1 ( γ ) , µ 2 ( γ ) , . . . , µ m ( γ )) = 0 for arbitrary γ ∈ GF ∗ ( q m ). Note that m X i =1 ν i f i,j ( µ 1 ( γ ) , µ 2 ( γ ) , . . . , µ m ( γ )) = m X i =1 ν i µ j ( β i γ ) = µ j m X i =1 ν i β i γ ! , where the last equ alit y is d ue to the linearit y of µ j in GF( q ). S ince γ ranges o ve r GF ∗ ( q m ), γ · P m i =1 ν i β i ranges o v er GF ∗ ( q m ) as well . How eve r, it is ob viously wr ong as it indicates that all elements in GF( q m ) are can b e represent ed by th e sub basis { β 1 ,. . . , β j − 1 , β j +1 , . . . , β m } . ( ii ). Sup p ose it is n ot true, sa y , m X i =1 ν i  f i,j ( µ 1 (1) , . . . , µ m (1)) , f i,j ( µ 1 ( γ ) , . . . , µ m ( γ )) , . . . , f i,j ( µ 1 ( γ N − 1 ) , . . . , µ m ( γ N − 1 ))  = 0 , 13 where ν i ∈ GF ( q ), for some γ ∈ GF( q m ) and j . Th is is equ iv alen t to that all elements P m i =1 ν i β i · g ( γ ) do n ot con tain the basis comp onent β j . Lemma 3 sho ws that P m i =1 ν i β i · g ( γ ) p ertains to a su b field. This indicates that γ = m X i =1 ν i β i ! − 1 · γ m X i =1 ν i β i ! , where th e t wo terms on the right side are the fir st and second elemen ts of P m i =1 ν i β i · g ( γ ), r esp ectiv ely . This indicates that γ is a subfield elemen t, whic h violates th e assumption. ✷✷ Note that Lemma 4.( ii ) ma y n ot hold tru e wh en γ b elongs to a sub fi eld of GF( q m ). E.g., let γ = α 5 b e in the field GF(2 4 ), then w e ha ve g ( α 5 ) = [1 , α 5 , α 10 , 1 , α 5 , α 10 , 1 , α 5 , α 10 , 1 , α 5 , α 10 , 1 , α 5 , α 10 ] lying in the subfield GF(2 2 ). W e observe that γ q s = m X i =1 µ q s i ( γ ) · β q s i = m X i =1 µ i ( γ ) · β q s i = m X i =1 β i m X j =1 µ ( j [ q s ]) i · µ j ( γ ) . (23) Com bining (23) and (16), w e obtain γ q s β i = m X j =1 β j m X k =1 µ ( i,k ) j m X l =1 µ ( l [ q s ]) k · µ l ( γ ) , (24) whic h immediately yields µ j ( γ q s β i ) = m X k =1 µ ( i,k ) j m X l =1 µ ( l [ q s ]) k · µ l ( γ ) . (25) Letting γ = β r , the ab o v e equalit y b ecomes µ j ( β q s r β i ) = m X k =1 µ ( i,k ) j m X l =1 µ ( l [ q s ]) k · µ l ( β r ) = m X k =1 µ ( i,k ) j µ ( r [ q s ]) k , (26) where b y definition µ l ( β r ) = 1 if l = r or 0 otherw ise. Consequ en tly , (25) can b e re-written as µ j ( γ q s β i ) = m X l =1 µ j ( β i β q s l ) · µ l ( γ ) . (27) It follo ws that µ j ( β i g ( γ q s )) = µ j ( β i β q s 1 ) · µ 1 ( g ( γ )) + µ j ( β i β q s 2 ) · µ 2 ( g ( γ )) + . . . + µ j ( β i β q s m ) · µ m ( g ( γ )) . (28) The follo wing t wo theorems c haracterize the intrinsic connection b et wee n subb asis and conjugate elemen ts. 14 Theorem 6 Given an e xp ande d gene r ator matrix G e ( γ , γ q s 1 , γ q s 2 , . . . , γ q s k − 1 ) , ( i ) . When γ is a non-subfield element in GF ∗ ( q m ) , the dimension of the subsp ac e sub c o de with r esp e ct to a subb asis { β i } m i =1 \{ β i 1 , β i 2 , . . . , β i t } i s mk − R ( Γ ) , (29) wher e R ( Γ ) denotes the maximum numb er of line arly indep e ndent r ows of the matrix Γ define d as Γ △ =                          µ i 1 ( β 1 β 1 ) . . . µ i 1 ( β 1 β m ) . . . . . . µ i t ( β 1 β 1 ) . . . µ i t ( β 1 β m ) . . . . . . . . . . . . . . . . . . . . . µ i 1 ( β m β 1 ) . . . µ i 1 ( β m β m ) . . . . . . µ i t ( β m β 1 ) . . . µ i t ( β m β m ) µ i 1 ( β 1 β q s 1 1 ) . . . µ i 1 ( β 1 β q s 1 m ) . . . . . . µ i t ( β 1 β q s 1 1 ) . . . µ i t ( β 1 β q s 1 m ) . . . . . . . . . . . . . . . . . . . . . µ i 1 ( β m β q s 1 1 ) . . . µ i 1 ( β m β q s 1 m ) . . . . . . µ i t ( β m β q s 1 1 ) . . . µ i t ( β m β q s 1 m ) . . . . . . . . . . . . . . . . . . . . . µ i 1 ( β 1 β q s k − 1 1 ) . . . µ i 1 ( β 1 β q s k − 1 m ) . . . . . . µ i t ( β 1 β q s k − 1 1 ) . . . µ i t ( β 1 β q s k − 1 m ) . . . . . . . . . . . . . . . . . . . . . µ i 1 ( β m β q s k − 1 1 ) . . . µ i 1 ( β m β q s k − 1 m ) . . . . . . µ i t ( β m β q s k − 1 1 ) . . . µ i t ( β m β q s k − 1 m )                          . ( 30) The dimension exhibits the lower b ound m ( k − t ) . ( ii ) . When γ is a pr op er subfield element in GF ∗ ( q m ) , such that GF ( q m γ ) c an b e r epr esente d by the minimal subb asis { β j 1 , β j 2 , . . . , β j r } (wher e ”minimial” me ans that any of its pr op er subset fails), the dimension of the subsp ac e sub c o de with r esp e ct to a subb asis { β i } m i =1 \{ β i 1 , β i 2 , . . . , β i t } i s mk − R ( Γ ′ ) , (31) wher e R ( Γ ′ ) denotes the maximum numb er of line arly indep e ndent r ows of the matrix Γ ′ define d as Γ ′ △ =                          µ i 1 ( β 1 β j 1 ) . . . µ i 1 ( β 1 β j r ) . . . . . . µ i t ( β 1 β j 1 ) . . . µ i t ( β 1 β j r ) . . . . . . . . . . . . . . . . . . . . . µ i 1 ( β m β j 1 ) . . . µ i 1 ( β m β j r ) . . . . . . µ i t ( β m β j 1 ) . . . µ i t ( β m β j r ) µ i 1 ( β 1 β q s 1 j 1 ) . . . µ i 1 ( β 1 β q s 1 j r ) . . . . . . µ i t ( β 1 β q s 1 j 1 ) . . . µ i t ( β 1 β q s 1 j r ) . . . . . . . . . . . . . . . . . . . . . µ i 1 ( β m β q s 1 j 1 ) . . . µ i 1 ( β m β q s 1 j r ) . . . . . . µ i t ( β m β q s 1 j 1 ) . . . µ i t ( β m β q s 1 j r ) . . . . . . . . . . . . . . . . . . . . . µ i 1 ( β 1 β q s k − 1 j 1 ) . . . µ i 1 ( β 1 β q s k − 1 j r ) . . . . . . µ i t ( β 1 β q s k − 1 j 1 ) . . . µ i t ( β 1 β q s k − 1 j r ) . . . . . . . . . . . . . . . . . . . . . µ i 1 ( β m β q s k − 1 j 1 ) . . . µ i 1 ( β m β q s k − 1 j r ) . . . . . . µ i t ( β m β q s k − 1 j 1 ) . . . µ i t ( β m β q s k − 1 j r )                          . ( 32) 15 Pr o of: ( i ). (28) indicates that        µ i 1 ( β ρ g ( γ q s l )) µ i 2 ( β ρ g ( γ q s l )) . . . µ i t ( β ρ g ( γ q s l ))        =        µ i 1 ( β ρ β s l 1 ) µ i 1 ( β ρ β s l 2 ) . . . µ i 1 ( β ρ β s l m ) µ i 2 ( β ρ β s l 1 ) µ i 2 ( β ρ β s l 2 ) . . . µ i 2 ( β ρ β s l m ) . . . . . . . . . . . . µ i t ( β ρ β s l 1 ) µ i t ( β ρ β s l 2 ) . . . µ i t ( β ρ β s l m )        ·        µ 1 ( g ( γ )) µ 2 ( g ( γ )) . . . µ m ( g ( γ ))        (33) where the matrix on the righ t side is exactly a folded row of Γ . The ab o v e equalit y immediately indicates that Γ corresp onds to th e co efficient ve ctor of null-subspace of the expand ed generator G e . Therefore, if a linear com b ination of ro ws of Γ results in an all-zero ro w, then the linear com bination with resp ect to G e results in a v alid subspace co deword. On the other hand, follo wing Lemma 4, a v alid sub space co deword exh ibits the all-zero co efficien t v ector asso ciated w ith null-subspace. The lo wer b ound is obtained by assuming the w orst-case that Γ is full-rank. ( ii ). Theorem 5 indicates that [ µ 1 ( g ( γ )) , µ 2 ( g ( γ )) , . . . , µ m ( g ( γ ))] = [ 0 , . . . , 0 , µ j 1 ( g ( γ )) , 0 , . . . , 0 , µ j 2 ( g ( γ )) , 0 , . . . , 0 , µ j r ( g ( γ )) , 0 , . . . , 0 ] . Th us, the equalit y (33) is reduced to        µ i 1 ( β ρ g ( γ q s l )) µ i 2 ( β ρ g ( γ q s l )) . . . µ i t ( β ρ g ( γ q s l ))        =        µ i 1 ( β ρ β s l j 1 ) µ i 1 ( β ρ β s l j 2 ) . . . µ i 1 ( β ρ β s l j r ) µ i 2 ( β ρ β s l j 1 ) µ i 2 ( β ρ β s l j 2 ) . . . µ i 2 ( β ρ β s l j r ) . . . . . . . . . . . . µ i t ( β ρ β s l j 1 ) µ i t ( β ρ β s l j 2 ) . . . µ i t ( β ρ β s l j r )        ·        µ j 1 ( g ( γ )) µ j 2 ( g ( γ )) . . . µ j r ( g ( γ ))        whic h concludes the part ( ii ). ✷✷ Next theorem sho ws a complemen tary view on the dimen s ion of th e subsp ace su b co des. Theorem 7 Given an exp ande d gener ator matrix G e ( γ , γ q s 1 , γ q s 2 , . . . , γ q s k − 1 ) , the dimension of the subsp ac e sub c o de with r esp e ct to a subb asis { β i 1 , β i 2 , . . . , β i t } i s m γ · ( t − R ( Θ )) , (34) wher e R ( Θ ) denotes the maximum numb er of line arly indep endent (define d in GF ( q m γ ) ) r ows of Θ define d as Θ △ =        β q m γ − z 1 i 1 β q m γ − z 2 i 1 . . . β q m γ − z κ i 1 β q m γ − z 1 i 2 β q m γ − z 2 i 2 . . . β q m γ − z κ i 2 . . . . . . . . . . . . β q m γ − z 1 i t β q m γ − z 2 i t . . . β q m γ − z κ i t        (35) wher e κ △ = m γ − k and { z 1 , z 2 , . . . , z κ } △ = { 1 , 2 , . . . , m γ − 1 }  { s 1 , s 2 , . . . , s k − 1 } . (36) 16 Pr o of: It is easily seen that any χ γ ( x ), χ γ q s 1 ( x ), . . . , χ γ q s k − 1 ( x ), where th e function χ is defi ned in (18 ), share κ = m γ − k common conjugacy ro ots, { γ − q z 1 , γ − q z 2 , . . . , γ − q z κ } . W e next define the p olynomial P ( x ) △ = β i 1 P 1 ( x ) + β i 2 P 2 ( x ) + . . . + β i t P t ( x ) , where P i ( x ) ∈ GF( q )[ x ], deg( P i ( x )) < m γ , i = 1 , 2 , . . . , t , and P ( γ − q z l ) = 0, l = 1 , 2 , . . . , κ . W e pro ceed to sh o w the one-to-one map b etw een a v alid co d ew ord p olynomial an d a p olynomial P ( x ) defined as ab o ve . Note that a v alid co dewo rd p olynomial c ( x ) can b e represen ted by c ( x ) = θ 0 g ( γ ) + θ 1 g ( γ q s 1 ) + . . . + θ k − 1 g ( γ q s k − 1 ) = x N − 1 p γ − 1 ( x )  θ 0 χ γ ( x ) + θ 1 χ γ q s 1 ( x ) + . . . + θ k − 1 χ γ q s k − 1 ( x )  , where θ i ∈ GF( q m ), i = 0 , 1 , 2 , . . . , k − 1. Clearly , P ( x ) = θ 0 χ γ ( x ) + θ 1 χ γ q s 1 ( x ) + . . . + θ k − 1 χ γ q s k − 1 ( x ) can b e rep resen ted by the subb asis { β i 1 , β i 2 , . . . , β i t } an d con tains the ro ots γ − q z 1 , γ − q z 2 , . . . , γ − q z κ . Con v ersely , n ote that a p olynomial is a v alid co deword p olynomial if it con tains the r o ots GF ∗ ( q m ) \ { γ − 1 , γ − q s 1 , . . . , γ − q s k − 1 } and can b e represente d by the su bbasis { β i 1 , β i 2 , . . . , β i t } . Clearly , x N − 1 p γ − 1 ( x ) P ( x ) is a v alid co dewo rd p olynomial. The condition that P ( x ) con tains the ro ots γ − q z 1 , γ − q z 2 , . . . , γ − q z κ indicates              0 = β i 1 P 1 ( γ − q z 1 ) + β i 2 P 2 ( γ − q z 1 ) + . . . + β i t P t ( γ − q z 1 ) 0 = β i 1 P 1 ( γ − q z 2 ) + β i 2 P 2 ( γ − q z 2 ) + . . . + β i t P t ( γ − q z 2 ) . . . 0 = β i 1 P 1 ( γ − q z κ ) + β i 2 P 2 ( γ − q z κ ) + . . . + β i t P t ( γ − q z κ ) W e observe that 0 =  β i 1 P 1 ( γ − q z l ) + β i 2 P 2 ( γ − q z l ) + . . . + β i t P t ( γ − q z l )  q m γ − z l = β q m γ − z l i 1 P 1 ( γ − 1 ) + β q m γ − z l i 2 P 2 ( γ − 1 ) + . . . + β q m γ − z l i t P t ( γ − 1 ) . Therefore, the preceding equation system can b e transf orm ed in to        β q m γ − z 1 i 1 β q m γ − z 1 i 2 . . . β q m γ − z 1 i t β q m γ − z 2 i 1 β q m γ − z 2 i 2 . . . β q m γ − z 2 i t . . . . . . . . . . . . β q m γ − z κ i 1 β q m γ − z κ i 2 . . . β q m γ − z κ i t               P 1 P 2 . . . P t        =        0 0 . . . 0        where P i denotes P i ( γ − 1 ). It follo ws th at the dimension of the solution sp ace is determined b y the matrix Θ . F urther n ote th at p i = P i ( γ − 1 ) wh ere p i ∈ GF( q m γ ) and deg ( P i ( x )) < m γ uniquely 17 determines th e p olynomial P i ( x ) ∈ GF( q )[ x ]. Finally , due to the homogeneousness of the ab o v e system, eac h in dep edent solution { p 1 , p 2 , . . . , p t } can b e arb itrarily scaled within GF ( q m γ ), and thus exhibits a dimension of m γ . Finally , it is w orth noting that the linear dep endence m ust b e in ligh t of GF( q m γ ), b ecause P i ( γ ) ∈ GF( q m γ ). The theorem follo ws. ✷✷ Corollary 4 Given an e xp ande d gene r ator matrix G e ( γ , γ q s 1 , γ q s 2 , . . . , γ q s k − 1 ) , ( i ) . ther e exist c o dewor ds with r esp e ct to a subb asis of m − k + 1 elements. ( ii ) . when k = m − 1 , no c o dewor d c an b e r epr e se nte d by a single-element su b b asis; when k = m − 2 , ther e exist c o dewor ds to b e r epr esente d by a two-element sub b asis { β 1 , β 2 } if and only if β △ = β 2 β 1 is a sub-field element and furthermor e, m β divides | z 2 − z 1 | , wher e z 1 , z 2 ar e as define d in (36) . ( iii ) . if a c o dewor d c is r e pr e se nte d by a minimal subb asis of i elements (her ein “minimal” me ans that any pr op er subset fails), then its weight is e q u al to i · q m γ − 1 ( q − 1) q m − 1 q m γ − 1 . Pr o of: P art ( i ) is due to that the n umb er of rows of − is m ( k − 1), and th us r esults in a p ositiv e dimension v alue. P art ( ii ) straight forwarly follo w s Theorem 7 (for the assertion of k = m − 2, it comes do wn to β q | z 2 − z 1 | = β ). ( iii ). W e observ e that the p olynomial x N − 1 p γ − 1 ( x ) is a common factor of { g γ q j ( x ) : 0 ≤ j < m } . T h us, µ l ( β i g ( γ q j )), 1 ≤ l, j, i ≤ m are co dewords asso ciated with the generator p olynomial x N − 1 p γ − 1 ( x ) . This also holds true for y that is a linear combination of the conjugacy set { β i g ( γ q j ) : 1 ≤ i ≤ m, 0 ≤ j < m } . Therefore, the conclusion follo ws Theorem 2. ✷✷ Note in the extreme case wh er e γ 1 , γ 2 , . . . , γ k comp ose a complete conjugacy class, G e ( γ 1 , γ 2 , . . . , γ k ) corresp onds to a BCH sub co d e, as explored in [20]. W e n o w p resen t examples in GF(2 8 ) to clarify the ab o v e theorem. g ( α 1 ) and g ( α 2 ) do not b elong to the conjugacy class of a nont rivial subfi eld, and thus can b e combined in a wa y to pr o duce eigh t linearly indep endent bin ary co dewo rds which are represente d by a subbasis with sev en elemen ts; w h ereas g ( α 1 ) and g ( α 16 ) comp ose the conjugacy class of the subfield GF(2 4 ), and thus can b e com b in ed in a w a y , u nder an appr opriate basis (sa y a comp osite basis { 1, α 17 , α 34 , α 51 , α 1 , α 18 , α 35 , α 52 } ), to p ro du ce eigh t linearly indep endent bin ary co dew ords wh ic h are represented b y a subb asis of four elemen ts (herein { 1, α 17 , α 34 , α 51 } , or { α 1 , α 18 , α 35 , α 52 } ). g ( α 1 ), g ( α 2 ) and g ( α 4 ) ma y b e com bined to p ro du ce co d ew ords that are represente d by a sub basis with six element s; g ( α 1 ), g ( α 2 ) and g ( α 16 ) may b e com bined to pro duce co d ew ords to b e represent ed b y a subbasis with four elemen ts; g ( α 1 ), g ( α 4 ), g ( α 16 ), and g ( α 64 ), un der an appropriate basis (sa y a comp osite basis { 1, α 17 , α 85 , α 102 , α 1 , α 18 , α 86 , α 103 } ), may b e combined to pr o duce co dew ords that are represented by a su bbasis with t wo elemen ts (herein { 1 , α 85 } , or { α 17 , α 102 } , or { α, α 86 } , or { α 18 , α 103 } ). W e pro ceed to establish the (n egativ e) r elation b et w een subb asis and n on-conjugate elemen ts. 18 Theorem 8 Given an exp ande d g e ner ator matrix G e ( γ 1 , γ 2 , . . . , γ k ) wher e γ i , i = 1 , 2, . . . , k , ar e non-subfield elements and satisfy φ ( γ i ) 6 = φ ( γ j ) , ∀ i 6 = j , no nontrivial c o dewor d c an b e r epr esente d by a pr op er subb asis. Pr o of: W e s ho w the correctness by con tradiction. Assum e there exists a nonzero co deword P k j =1 θ j g ( γ j ) (where θ j ∈ GF( q m )) which can b e represent ed by a prop er su b basis, sa y β l not b eing included. It is sho wn in Theorem 5 th at th e co deword w ith only one n ontrivial co efficient θ j can not b e represen ted b y a subbasis. W e pro ceed to consider the remaining cases where at least t w o co efficients are non triv- ial. Without loss of generalit y , w e assu me θ 1 and θ 2 are nontrivial. Recall that (as sho wn in (19)) the p olynomial µ l ( θ 1 g γ 1 ( x )) is n ot divisible by p γ − 1 1 ( x ), wh ereas the all other p olynomials µ l ( θ i g γ i ( x )), i = 2 , 3 , . . . k , are all divisible by p γ − 1 1 ( x ). Therefore, θ 1 m ust b e trivial. T h e theorem f ollo ws. ✷✷ In [10], a explicit f orm ula u tilizing du al-basis is giv en for determining the dimension of subspace sub co d es defined in GF (2 m ). C learly , Theorems 6, 7 and 8 rev eal an alternativ e and more general in terpretation on the dimension of su bspace sub co des, and particularly reve al th at the dimen sion of a supspace sub co de can b e optimized through choosing appropr iate comp osite basis. W e next presen t examples to clarify the ab o v e assertion. When a 6-dimensional sub space in GF (2 8 ) is considered, w e maximize the sub co de dim en sion by emplo ying the comp osite basis { 1 , α 17 , α 85 , α 102 , α 1 , α 18 , α 86 , α 103 } and su bsequently choosing the subb asis { 1 , α 17 , α 85 , α 102 , α 1 , α 86 } . Under this su bspace, G e ( α, α 4 ) con tains a sub co d e of eigh t dimensions (recall that α and α 4 p ertain to th e same conjugacy class of the subfield GF(2 2 )), whereas und er a regular p olynomial basis it d o esnot con tain a 6-dimensional subspace sub co de. G e ( α 17 ) con tains a su b co de w ith dimension 4 (recall that g ( α 17 ), α 17 g ( α 17 ), α 85 g ( α 17 ), α 102 g ( α 17 ) can b e represen ted by the subb asis { 1 , α 17 , α 85 , α 102 } . G e ( α 85 ) cont ains a sub- co de with d imension 6 (recall that g ( α 85 ) and α 85 g ( α 85 ) can b e represen ted by the subb asis { 1 , α 85 } ; α 17 g ( α 85 ) and α 102 g ( α 85 ) can b e repr esen ted by the su bbasis { α 17 , α 102 } ; α g ( α 85 ) and α 86 g ( α 85 ) can b e represented by the su bbasis { α, α 86 } ). The follo win g corollary is an extension of Corollary 3. Corollary 5 Given an exp ande d gener ator matrix G e ( γ 1 , γ 2 , . . . , γ k ) , let l b e the minimum numb er of b asis elements to r epr esent any p articular c o dewor d gener ate d by G e ( φ ( γ ) ∩ { γ 1 , γ 2 , . . . , γ k } ) , then p γ − 1 ( x ) divi des either al l m c omp onent p olynomial s or up to m − l c omp onent p olynomials of any c o dewor d p olynomial gener ate d by G e ( γ 1 , γ 2 , . . . , γ k ) . Pr o of: If a co deword is generated b y G e ( { γ 1 , γ 2 , . . . , γ k }\ φ ( γ )), then p γ − 1 ( x ) d ivides all m comp onent p olynomials, as sho wn in Corollary 3 . Otherwise, we divides th e co deword c into t w o parts c = c 1 + c 2 , where c 1 is generated by G e ( { γ 1 , γ 2 , . . . , γ k }\ φ ( γ )) and c 2 is generated by G e ( φ ( γ ) ∩ { γ 1 , γ 2 , . . . , γ k } ). W e n ote that p γ − 1 ( x ) d ivides all comp onen t co d ew ord p olynomials of c 1 , while d ivides only the zero comp onent p olynomials of c 2 . T he corollary follo w s. ✷✷ 19 The follo win g corollary charact erizes the num b er of linearly indep endent comp onen ts of a co dew ord. Corollary 6 L et a c o dewor d c b e c omp ose d of c = θ 1 g ( γ 1 ) + θ 2 g ( γ 2 ) + . . . + θ k g ( γ k ) , θ i 6 = 0 , ∀ i, and l i b e the minimum size of subb asis to r epr esent any p articular c o dewor d gener ate d by G e ( φ ( γ i ) ∩ { γ 1 , γ 2 , . . . , γ k } ) under a giv en b asis { β 1 , β 2 , . . . , β m } . Then, the numb er of line arly indep endent c om- p onent c o dewor ds of c is at le ast max { l 1 , l 2 , . . . , l k } . Pr o of: Without loss of generalit y , let l 1 b e the largest, i.e., l 1 ≥ l i , i = 2, 3, . . . , k . W e first decomp ose th e co d ew ord c in to t w o p arts, c = c 1 + c 2 , such that c 1 corresp onds to the generator m atrix G e ( φ ( γ 1 ) ∩ { γ 1 , γ 2 , . . . , γ k } ), and c 2 corresp onds to the generator matrix G e ( { γ 1 , γ 2 , . . . , γ k }\ φ ( γ 1 )). W e first show that the num b er of linearly ind ep endent comp onent co dewo rds of c 1 is at least l 1 through con tradiction. Without loss of generalit y , we assume that the component co dewo rds, µ l ( c 1 ), µ l +1 ( c 1 ), . . . , µ m ( c 1 ), are linearly dep endent on the linearly indep endent comp onen t co dewords, µ 1 ( c 1 ), µ 2 ( c 1 ), . . . , µ l − 1 ( c 1 ) ( l ≤ l 1 ), suc h that, µ i ( c 1 ) = ν i, 1 µ 1 ( c 1 ) + ν i, 2 µ 2 ( c 1 ) + . . . + ν i,l − 1 µ l − 1 ( c 1 ) , i = l , l + 1 , . . . , m. Consequent ly , we obtain c 1 = l − 1 X i =1 β i µ i ( c 1 ) + m X i = l β i µ i ( c 1 ) = l − 1 X i =1 β i µ i ( c 1 ) + m X i = l β i l − 1 X j =1 ν i,j µ j ( c 1 ) = l − 1 X i =1 µ i ( c 1 )( β i + m X j = l ν j,i β j ) . The ab o v e equalit y in dicates that c 1 can b e represen ted by the sub basis ( β 1 + m X i = l ν i, 1 β i , β 2 + m X i = l ν i, 2 β i , . . . , β l − 1 + m X i = l ν i,l − 1 β i ) , whic h has l − 1 ≤ l 1 − 1 elemen ts (and can b e expand ed to an alternativ e basis). This clearly violates the definition of l 1 . On the other hand, we recall that p γ − 1 1 ( x ) divides all comp onent p olynomials of c 2 , whereas it divides only all-zero comp onent p olynomials of c 1 . Therefore, add ing c 2 to c 1 cannot redu ce the n umber of linearly ind ep endent comp onent co dewo rds of c 1 . W e th us conclude the corollary . ✷✷ In [18], a low er b ound on the minimum distance of expanded cyclic co des is obtained by treating it as a generalized concatenated co de. The follo wing th eorem establishes an impro ve d b ound by incorp orating the preceding new insigh ts. 20 Theorem 9 Given an exp ande d g e ner ator matrix G e ( γ 1 , γ 2 , . . . , γ k ) , the minimum distanc e is b ounde d by d min ≥ min 1 ≤ i ≤ m { i · d ( i ) } , (37) wher e d ( i ) denotes the minimum distanc e of the sub c o de asso ciate d with the gener ator p olynomial G ( i ) ( x ) = x N − 1 LCM { p γ − 1 i ( x ) : φ ( γ i ) ∩ { γ 1 , γ 2 , . . . , γ k } r esults in a subb asis with up to i elements } (38) wher e LCM stands for “L e ast Common Multiplier”. In essence, in [18], the minimum d istance of an outer co de is sh o wn to b e b ou n ded b y the largest n umber of consecutiv e conju gate elements, wh er eas it is p recisely computed thr ough Theorems 6, 7, 8. W e present th ree examples in GF(2 5 ) to sh ed ligh t on the pr op osed b oun d in con trast to the b ound in [18]. Giv en the generator matrix G e ( α 21 , α 22 ), where α 22 = α 21 × 4 , the p rop osed lo w er b ound is computed as 16 × 4 = 64 , wh ereas the b oun d pro vided in [18] is 48. Give n the generator matrix G e ( α 21 , α 22 , α 23 ), th e prop osed lo w er b ound is min { 16 × 4 , 12 × 5 } = 60 , whereas the b ound pro vided in [18] is 48. Giv en the generator matrix G e ( α 18 , α 19 , α 20 , α 21 , α 22 ), where α 20 = α 18 × 8 and α 22 = α 21 × 4 , the pr op osed lo w er b ound is min { 10 × 4 , 8 × 5 } = 40 , whereas th e b oun d pr ovided in [18] is 36. It is worth noting that the prop osed b ound is rather lo ose for h igh rate co des. F or instance, let the co de r ate of a Reed-Solomon co de in GF(2 m ) b e greater than one h alf, then, when m is a p rime, the p r op osed b ound on the minim um distance of the resulting expand ed co de r educes trivially to 2 m , as G ( m ) ( x ) = x − 1 and subsequently d ( m ) = 2 (actually the w orse case is that G ( m ) ( x ) = 1 and d ( m ) = 1); alternativ ely , when m is not a pr ime, G ( m ) ( x ) m a y con tain the min imal p olynomials of subfield elemen ts, and thus th e b ound can b e somewhat improv ed. The follo w in g theorem s h o ws that the binary expanded Reed-Solomon cod es, regardless of r ealiza- tion basis, are asymptotically bad, in con trary to th e prev alen t conjecture (cf. [9]), as we ll as to the ensem ble of generalized Reed-Solomon co des whic h asymptotically achiev es the Gilb ert-V arsh amov b ound [15]. Theorem 10 F or a se quenc e of primitive (2 m − 1 , ⌊ (2 m − 1) ⌋ ) R e e d-Solomon c o des with a fixe d r ate r and a fixe d starting sp e ctrum δ (i.e., its gener ator p olynomial is define d as G ( x ) = Q δ − 1+ ⌈ (1 − r )(2 m − 1) ⌉ i = δ ( x − α i ) ), their b inary H amming minimum distanc es d min satisfy lim m →∞ d min m (2 m − 1) = 0 . (39) Pr o of: W e first consider the case δ ≥ 0. Let k = ⌊ log 2 ((2 m − 1) r − δ ) ⌋ . 21 W e obser ve that the set { β i g ( α q s ) : 1 ≤ i ≤ m, 0 ≤ s ≤ k } is con tained in the b inary expansion of the generator matrix. I n accordance w ith T heorem 6, there exists a co dewo rd that is repr esen ted by a subbasis with up to m − k elemen ts and is with we igh t at most ( m − k )2 m − 1 . Therefore, lim m →∞ d min m (2 m − 1) ≤ lim m →∞ ( m − k )2 m − 1 m (2 m − 1) ≤ lim m →∞ ( m − log 2 ( r 2 m − r − δ ) + 1)2 m − 1 m (2 m − 1) = 0 . No w we consider the alternativ e case δ < 0. Let k 1 = ⌈ log 2 ( − δ ) ⌉ , k 2 = ⌊ log 2 ((2 m − 1) r − δ ) ⌋ . F ollo wing Corollary 4 .( i ), there exists a co dewo rd that is represente d b y a subb asis with u p to m − ( k 2 + k 1 ) elemen ts and is with we ight at most ( m − k 2 + k 1 )2 m − 1 . Again, w e hav e lim m →∞ d min m (2 m − 1) ≤ lim m →∞ ( m − k 2 + k 1 )2 m − 1 m (2 m − 1) ≤ lim m →∞ ( m − log 2 ( r 2 m − r − δ ) + ⌈ log 2 ( − δ ) ⌉ + 1)2 m − 1 m (2 m − 1) = 0 . The pro of is completed. ✷✷ In addition, elemen ts of ve ry s m all s ubfield also con tribute to lo w weigh t. E.g., when m is ev en, w  g ( α N/ 3 )  = 2 × 2 2 − 1 2 m − 1 2 2 − 1 = 4 N 3 under an appr op r iate comp osite b asis (sa y { 1 , α 1 , . . . , α m/ 2 − 1 , α N/ 3 , α N/ 3+1 , . . . , α N/ 3+ m/ 2 − 1 } ), w h ere α N/ 3 p ertains to the subfield GF(2 2 ). V. Concluding Remarks The pap er has a threefold pu rp ose. Th e fi rst pu rp ose is to present an explicit description of expan d ed cyclic co d es defined in GF( q m ). Th e second purp ose of this pap er is to identify a class of constan t- w eigh t cyclic co d es wh ic h ac hiev e the Plotkin b ou n d. The last pur p ose of the pap er is to c haracterize expanded cyclic co des utilizing the prop osed expanded generator matrix and parit y chec k matrix. W e c haracterize the p r op erties of comp onen t cod ew ords of a cod ew ord and particularly iden tify the pr ecise conditions under which a co deword can b e represen ted by a su bbasis. 22 Our analysis seems to su ggest that s ym b ol-wise minimum w eigh t co dewords are irrelev ant to the bit-wise minimum weigh t co dewords. Our extensiv e simulations s uggest that the comp onent co de- w ords corresp ondin g to d ifferent indices m ay not reac h (close to) minim um w eigh t simultaneo usly and subsequently the pr op osed the minimum distance b ound is rather lo ose (for instance, w hen the co de rate of a Reed-Solomon co de in GF(2 m ) is greater than half, the prop osed b ound on the minim um distance of the resulting exp anded co d e b y and large reduces to 2 m ). Th erefore, it is imp erative to determine a substanti ally tigh ter b ound . Moreo ver, w e strongly b elieve that this is also critical to ex- plicitly find “go o d” co des fr om bin ary expanded cyclic co des (without generalization, whic h inevitably renders the analysis int ractable). Ac kno wledgemen t The author w ould lik e to thank Dr. Jun Ma, Prof. Jorn Justesen, and particularly , P rof. Marc F ossorier, for many constructiv e commen ts on impr oving the p r esen tation of the man uscript. References [1] N. Q. A, L. Gyorfi, and J. 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