Construction of Z4-linear Reed-Muller codes

Construction of Z 4 -linear Reed-Muller co des ∗ J. Pujol, J. Rif` a, F. I. Solo v’ev a † Abstract New quaternary Plotkin constructions are giv en and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension and the minimum distance are studied. Using these constructions new families of quaternary Reed-Muller codes are b uilt with t he p eculiarit y that after using th e Gray map the obtained Z 4 -linear codes hav e the same parameters an d fund amen tal properties as the cod es in the usual binary linear Reed-Muller family . T o make more evident the du alit y relationships in the co nstru cted famili es t he co ncept of Kroneck er inner produ ct is introd uced. 1 In tro duction In [1 3] Nechaev int r o duced the concept of Z 4 -linearity of binar y co des and later, in [7], Hammons, Kumar, Calderba nk, Sloane and Sol´ e sho wed that several families of bina r y co des are Z 4 -linear. In [7] it is prov ed that the binary linear Reed-Muller co de RM ( r , m ) is Z 4 -linear for r = 0 , 1 , 2 , m − 1 , m and is not Z 4 -linear for r = m − 2 ( m ≥ 5). In a subsequent w or k, Ho u, Lahtonen a nd Kop onen [8], pro ved that R M ( r , m ) is not Z 4 -linear for 3 ≤ r ≤ m − 2. In [7] it is in tro duced a construction of co des, called QRM ( r , m ), base d on Z 4 -linear co des, such that after doing mo dulo tw o we obtain the usual binary linear Reed-Muller ( RM ) co des. In [2, 3] such family of co des is studied and their parameters are computed as well as the dimension of the kernel and rank . In [16] a kind of Plotkin construction was use d to build a family of additive Reed-Muller co des and also in [1 9] it was used a Plo tkin construction to obtain a sequence of quaterna ry linear Reed-Muller like co des. In b oth last quo ted constructions, the images of the o btained co des under the Gray map are binary co des with the same parameters as th e bina r y linea r RM co des. Moreover, on the o ther hand, in [5 , 9, 10, 1 4] were cla s sified all the non-equiv alent Z 2 Z 4 -linear extended 1-p erfect codes a nd t heir duals, the Z 2 Z 4 -linear Hadamard co des. It ∗ This work has b een partially supp orted by the Spanish MEC and the Europ ean FEDER Gran t MTM2006-03250. Pa rt of the m ate ri al in Section II I of this paper was pr esented at the 17th Symposium on App li ed algebra, Algeb rai c algorithms, and Error Correcting Co des (AAECC), Bangalore, India, Decem b er 2007. † J. Pujol and J. Rif` a are wi th the Departmen t of Information and Comm unications Engi- neering, Universitat Aut` ono ma de Barcelona, 08193-Bellaterra, Spain. F. I. Solo v’ev a is with the Sobolev Institute of Mathematics and No vosibirsk State Universit y , Nov osibir sk, Russia. 1 is a natural q ue s tion to a sk for the existence of families of quaternar y linear co des such that, after the Gray map, the cor respo nding Z 4 -linear co des have the same parameters as t he well known family of binary linear RM co des. In thes e new families, like in the usual R M ( r , m ) family , the co de with ( r, m ) = (1 , m ) should b e a Hadamar d code and the co de with ( r, m ) = ( m − 2 , m ) should b e an extended 1-p erfect co de. It is well known that an easy wa y to built the binary R M family of co des is by using the Plotkin c o nstruction [12]. So, it seems a g oo d matter of study to try to g eneralize the Plo tkin cons tr uction to the quaternar y linear codes and try to obtain new families of co des whic h con tain the ab o ve men tioned Z 2 Z 4 - linear Hadamard co des and Z 2 Z 4 -linear extended 1-p erfect co des and fulfill the same pr oper ties from a parameters po in t of view ( leng th, dimension, minim um distance, inclus io n and dualit y r elationship) than the binary RM family . In this paper w e begin by studying the Z 4 -linear cas e a nd we org anize it in the following way . In Section 2 we in tro duce the concept of quaternary co de and g iv e some constructions that c o uld be seen as quaternary genera liz a tions of the well known binary Plotkin constructio n. In Section 3, w e co nstruct several families of Z 4 -linear Reed-Muller codes and prov e that they have similar parameters as the classical binary RM co des but they are not lin ea r. In Section 4 , w e discuss the concept of dua lit y fo r the constructed Z 4 -linear Reed-Muller codes a nd, finally , in S ectio n 5 we give some c onclusions and further re s earch in the topic. 2 Constructions of quaternary co des 2.1 Quaternary co des Let Z 2 and Z 4 be the ring of integers modulo t wo a nd modulo four, resp ectiv ely . Let Z n 2 be the set of a ll bina ry vectors of length n and Z N 4 be the set of all quaternary vectors of length N . An y non-empty subse t C of Z n 2 is a binary co de and a subgr oup of Z n 2 is called a binary line ar c o de . Equiv alently , any non-empty subset C o f Z N 4 is a quater nary code and a subgr oup o f Z N 4 is called a quaternary line ar c o de . In g e neral, a n y non-empt y subgro up C of Z α 2 × Z β 4 is an additive c o de . The H a mming w eight w H ( u ) o f a v ector in Z n 2 is the n umber of its no nzero co ordinates. The Hamming distance d ( u , v ) b et ween tw o vectors u , v ∈ Z n 2 is d ( u , v ) = w H ( u − v ). F or quaterna ry co des it is mor e appropria te to use the Lee metric [11]. In Z 2 the Lee weight coincides with the Ha mming weigh t, but in Z 4 the Le e weigh t of their elements is w L (0) = 0 , w L (1) = w L (3) = 1, and w L (2) = 2. The Lee w eight w L ( u ) of a vector in Z N 4 is the addition of the Lee weigh t of all the co ordinates. The Lee distance d L ( u , v ) b et ween tw o vectors u , v ∈ Z N 4 is d L ( u , v ) = w L ( u − v ). Let C be an add itive c o de , s o a subgroup o f Z α 2 × Z β 4 and let C = Φ( C ), where Φ : Z α 2 × Z β 4 − → Z n 2 , n = α + 2 β , is g iv en by Φ( u , v ) = ( u , φ ( v )) for any u from Z α 2 and any v fr o m Z β 4 , where φ : Z β 4 − → Z 2 β 2 is the usual Gray map, so φ ( v 1 , . . . , v β ) = ( ϕ ( v 1 ) , . . . , ϕ ( v β )) , and ϕ (0) = (0 , 0) , ϕ (1) = (0 , 1) , ϕ (2) = 2 (1 , 1 ), ϕ (3) = (1 , 0). W e will use the symbols 0 , 1 and 2 for the all zeroes, the all ones and the all tw os v ectors, resp ectiv ely (b y the cont ext it will be alwa ys clear we speak about the bina r y v ecto r s 0 , 1 o r quaternary , it will a lso be c lear the le ng th of th e vectors). Hamming and Lee weigh ts, a s well as Hamming and Lee distances, can b e generalized, in a na tural way , to vectors in Z α 2 × Z β 4 by adding the cor r esponding weigh ts (or distances) of the Z α 2 part a nd the Z β 4 part. Since C is a subgroup of Z α 2 × Z β 4 , it is also is omorphic to an ab elian s tr ucture like Z γ 2 × Z δ 4 . Therefor e, we have that |C | = 2 γ 4 δ and the n um b er of o rder t w o co dew or ds in C is 2 γ + δ . W e ca ll suc h co de C an additive c o de of typ e ( α, β ; γ , δ ) and the binary imag e C = Φ( C ) a Z 2 Z 4 -line ar c o de of typ e ( α, β ; γ , δ ). In the sp ecific case α = 0 we see that C is a quaternary linear co de and its binary image is ca lled a Z 4 -line ar c o de . Note that the binary length o f the binary co de C = Φ( C ) is n = α + 2 β . The minim um Hamming distance d of a Z 2 Z 4 -linear c o de C is the minimum v alue of d ( u , v ), where u , v ∈ C and u 6 = v . Notice that the Hamming distance of a Z 2 Z 4 -linear co de C co incides with the Lee distance defined in the additive co de C = Φ − 1 ( C ). F ro m now on, when we work with distances it must b e understo o d that we deal with Hamming dista nces in the cas e of binar y co des o r Lee distanc e s in th e case o f additive co des. Although C could not have a basis , it is appr o priate to define a generator matrix fo r C as G =  B 2 Q 2 B 1 Q 1  , where B 2 is a γ × α matrix; Q 2 is a γ × β matrix; B 1 is a δ × α matrix and Q 1 is a δ × β ma trix. Matrices B 1 , B 2 are binary and Q 1 , Q 2 are quater nary , but the e ntries in Q 2 are only zero es or t wos. Two additive co des C 1 and C 2 bo th of the same length are said to b e monomi- al ly e quivalent , if one can be obtained from the other by per m uting the coor di- nates and m ultiplying b y − 1 of c e rtain co ordinates. Additiv e co des which differ only by a permutation o f co ordinates are said to be p ermutation e quivalent . F or Z 2 Z 4 -linear co des is usual to use the following definition of inner pr oduct in Z α 2 × Z β 4 that we will call the standar d inner pr o duct [1 8, 4]: h u , v i = 2( α X i =1 u i v i ) + α + β X j = α +1 u j v j ∈ Z 4 , (1) where u , v ∈ Z α 2 × Z β 4 . W e can also write the standard inner product as h u , v i = u · J N · v t , where J N =  2 I α 0 0 I β  , N = α + β , is a diago nal ma trix over Z 4 . Note that when α = 0 the inner pro duct is the usual one for vectors over Z 4 and when β = 0 it is t wice the usua l one fo r v ecto r s ov er Z 2 . 3 F or α = 0 and N = β = 2 i , i = 1 , 2 , 3 , . . . , w e can define the inner pro duct in an alternative way . Let K 2 =  1 0 0 3  be a matrix over Z 4 and define K N = N log 2 ( N ) j =1 K 2 where N denotes the K r onec ker pro duct of matrices. W e call the Kr one cker inner pr o duct the following: h u , v i ⊗ N = u · K N · v t . (2) The add itive dual c o de of C , denoted b y C ⊥ , is defined in the standard way as C ⊥ = { u ∈ Z α 2 × Z β 4 | h u , v i = 0 for a ll v ∈ C } or, using the K roneck er inner pr oduct C ⊥ = { u ∈ Z α 2 × Z β 4 | h u , v i ⊗ N = 0 for all v ∈ C } . The definition and notatio ns will b e the same for the Z 4 -duality o btained by using the s tandard inner pr o duct o r the Kro nec ker inner pro duct and the difference will be clear f r o m the con text. Note that h u , v i ⊗ N = u · K N · v t = h u , v · K N i . Hence, b oth additive dua l co des by using the s ta ndard inner pro duct or the Kronecker inner pro duct, resp ectiv ely , ar e monomia lly equiv alent and so they ha ve t he same w eight dis- tribution. F or b oth inner pro ducts, the additive dual co de C ⊥ is also an additive co de, that is a subgroup of Z α 2 × Z β 4 . Its weigh t enumerator p olynomial is related to the w eig h t enumerator po lynomial of C by the MacWilliams identit y [6]. The corres p onding binary co de Φ( C ⊥ ) is denoted by C ⊥ and called the Z 2 Z 4 -dual c o de o f C . In the case α = 0, the co de C ⊥ is als o called the quaternary dual c o de of C and C ⊥ the Z 4 -dual c o de of C . Notice that C and C ⊥ are not dual in the binary linear sense but the w eig h t enumerator polyno mial of C ⊥ is the McWilliams transfor m of the weigh t enumerator poly nomial o f C . Given an additive co de C of type ( α, β , γ , δ ) it is known the type of the a dditiv e dual code ([4] for additiv e code s with α 6 = 0 and [7] for additiv e codes with α = 0). In the prese nt pap er, a s we w ill see later, the duality concept using the Kronecker inner pro duct will make more visible the prop erty that if a co de C belo ngs to a family of Reed-Muller co des then its dual c o de belong s to the same family . F ro m now o n, we fo cus o ur a tten tion sp ecifically to additive co des with α = 0, so quaterna ry line a r co des suc h that after the Gray map they giv e rise to Z 4 -linear co des. Given a quaternary linear co de of type (0 , β ; γ , δ ), we will write ( N ; γ , δ ) to say that α = 0 and β = N . 2.2 The Plotkin constr uction In this section, we show that the w ell-known binary Plotkin construction can be generalized t o quaternary linear co des. Let A and B b e t w o quater nary linea r co des of types ( N ; γ A , δ A ) and ( N ; γ B , δ B ) and minim um distances d A , d B , resp ectiv ely . Given u ∈ Z N 4 define supp ( u ) ⊂ { 1 , . . . , N } as the set of nonzero coor dina tes of vector u . 4 Definition 1 (Plotkin Construction) Given two qu atern ary line ar c o des A and B , we d efine a quaternary line ar c o de as P C ( A , B ) = { ( u 1 | u 1 + u 2 ) : u 1 ∈ A , u 2 ∈ B } . It is easy to see that if G A and G B are g enerator matrices of A and B , resp ectively , then the matrix G P C =  G A G A 0 G B  is a generator matr ix of the co de P C ( A , B ). Prop osition 2 The quaternary l ine ar c o de P C ( A , B ) define d using the Plotkin c onst ruction is of typ e (2 N ; γ , δ ) , wher e γ = γ A + γ B and δ = δ A + δ B ; t he binary length is n = 4 N ; the size is 2 γ +2 δ and the minimum distanc e is d = min { 2 d A , d B } . Pr o of: The type, the binary length and the size of P C ( A , B ) can b e easily computed fro m the definition o f the code. The minim um distance c a n b e es tab- lished a s in the binary case [12] but, by completeness, we include the proo f. Let us consider an y v ecto r u ∈ P C ( A , B ) suc h that u = ( u 1 | u 1 + u 2 ), where u 1 ∈ A and u 2 ∈ B . Since P C ( A , B ) is a quaterna ry linear co de, it is enough to prov e that t he w eight w L ( u ) is not less than d . If u 2 = 0 , then w L ( u ) = 2w L ( u 1 ) ≥ 2 d A . If u 2 6 = 0 , b y using the triang le inequality we immediately o btain w L ( u ) = w L ( u 1 ) + w L ( u 1 + u 2 ) ≥ w L ( u 2 ) ≥ d B . Hence d ≥ min { 2 d A , d B } . The equality holds b ecause taking the sp ecific vectors u 1 ∈ A wit h minim um weight d A and u 2 ∈ B with minimum weigh t d B we obtain w L ( u 1 | u 1 ) = 2 d A and w L ( 0 | v 2 ) = d B .  2.3 The quaternary P lotkin construction A use ful genera lization o f the ab o ve construction to obtain quaternary linear co des is the following constr uction, calle d the quaternary Plotkin c onstr u ction . Such cons truction was used, for example, in [10] for the classificatio n of all Z 4 -linear Hadamard co des. Definition 3 (Quaternary Plotkin Construction) Given two qu aternary lin- e ar c o des A and B , we define the quaternary li ne ar c o de QP ( A , B ) = { ( u 1 | u 1 + u 2 | u 1 + 2 u 2 | u 1 + 3 u 2 ) : u 1 ∈ A , u 2 ∈ B } . It is eas y to see that if G A and G B are generator matrices of A and B , then the ma trix G QP =  G A G A G A G A 0 G B 2 G B 3 G B  is a generator matr ix of the co de QP ( A , B ). 5 Prop osition 4 The quaternary line ar c o de QP ( A , B ) given in Definition 3 is of typ e (4 N ; γ , δ ) , wher e γ = γ A + γ B and δ = δ A + δ B ; the binary length is n = 8 N ; the size is 2 γ +2 δ and the minimum distanc e is d ≥ min { 4 d A , 2 d B } . Pr o of: The t yp e, the bina ry length and the size of Q P ( A , B ) can b e easily computed from the definition of the code. T o chec k the minimum distance of QP ( A , B ) let us cons ider any vector u ∈ QP ( A , B ). V ector u can b e repre s en ted by u = ( u 1 | u 1 | u 1 | u 1 ) + ( 0 | u 2 | 2 u 2 | 3 u 2 ), wher e u 1 ∈ A and u 2 ∈ B . Since QP ( A , B ) is a quaterna r y linea r code it is enough to sho w that t he w eight of u is at least d . If u 2 = 0 , then w L ( u ) = 4w L ( u 1 ) ≥ 4 d A . The equality holds taking a vector u 1 ∈ A of minimu m w eight. F or u 2 6 = 0 w e have w L ( u ) = w L ( u 1 | u 1 + u 2 | u 1 + 2 u 2 | u 1 + 3 u 2 ) = (w L ( u 1 ) + w L ( u 1 + u 2 )) + (w L ( u 1 + 2 u 2 ) + w L ( u 1 + 2 u 2 + u 2 )) ≥ w L ( u 2 ) + w L ( u 2 ) (by using t he triangle inequalit y) ≥ 2 d B .  The Plotkin a nd the quaternar y Plotkin constructions can b e c om bined in a double Plotkin c onstru ction . Let A , B , C and D be four quaternar y linear co des of types ( N ; γ A , δ A ), ( N ; γ B , δ B ), ( N ; γ C , δ C ), and ( N ; γ D , δ D ) a nd minim um distances d A , d B , d C , d D , res p ectively . Definition 5 (Double Plotkin Construction) Given A , B , C and D four quaternary line ar c o des, we define the quaternary line ar c o de DP ( A , B , C , D ) = { ( u 1 | u 1 + u 2 | u 1 +2 u 2 + u 3 | u 1 +3 u 2 + u 3 + u 4 ) : u 1 ∈ A , u 2 ∈ B , u 3 ∈ C , u 4 ∈ D } . It is easy to see that if G A , G B , G C and G D are generator matrices of A , B , C and D , then the matrix G DP =     G A G A G A G A 0 G B 2 G B 3 G B 0 0 G C G C 0 0 0 G D     is a generator matr ix of the co de D P ( A , B , C , D ). Prop osition 6 The quaternary line ar c o de D P ( A , B , C , D ) given in Defin i- tion 5 is of typ e (4 N ; γ , δ ) , wher e γ = γ A + γ B + γ C + γ D and δ = δ A + δ B + δ C + δ D ; the binary length is n = 8 N ; the size is 2 γ +2 δ and t he minimum distanc e is d ≥ min { 4 d A , 2 d B , 2 d C , d D } . 6 Pr o of: The t yp e, the binar y length and the size of t he co de DP ( A , B , C , D ) can b e easily computed from the definition. T o check the minim um dista nce of the co de D P ( A , B , C , D ) let us consider any vector u fr om this co de. It can b e r epresent ed as u = ( u 1 | u 1 | u 1 | u 1 ) + ( 0 | u 2 | 2 u 2 | 3 u 2 ) + ( 0 | 0 | u 3 | u 3 ) + ( 0 | 0 | 0 | u 4 ), where u 1 ∈ A , u 2 ∈ B , u 3 ∈ C and u 4 ∈ D . Since DP ( A , B , C , D ) is a q uaternary linear co de it is enough to show that t he w eight of u is, a t least, d . If u 2 = 0 then we can write u = ( u 1 | u 1 | u 1 | u 1 ) + ( 0 | 0 | u 3 | u 3 + u 4 ) s o that u ∈ P C (( A|A ) , P C ( C , D )) , where ( A|A ) is the co de ge nerated by ( G A |G A ). Using Prop osition 2 w e o btain w L ( u ) = min { 2 d ( A|A ) , d P ( C , D ) } = min { 4 d A , min { 2 d C , d D }} = min { 4 d A , 2 d C , d D } . If u 2 6 = 0 then we distinguish two cases. If u 4 = 0 then w L ( u ) = w L ( u 1 | u 1 + u 2 ) + w L ( u 1 + 2 u 2 + u 3 | u 1 + 3 u 2 + u 3 ) ≥ w L ( u 2 ) + w L ( u 2 ) ≥ 2 d B using twice the tr iangle inequality . If u 4 6 = 0 then w L ( u ) = w L ( u 1 | u 1 + u 2 ) + w L ( u 1 + 2 u 2 + u 3 | u 1 + 3 u 2 + u 3 + u 4 ) ≥ w L ( u 2 ) + w L ( u 2 + u 4 ) ≥ w L ( u 4 ) ≥ d D .  Note tha t in case B = C the bound is tight b ecause d B = d C and the minimum distance d = min { 4 d A , 2 d C , d D } c a n b e obtained tak ing sp ecific v ectors from A , C or D . 2.4 The BQ-Plotkin constr uction W e slig h tly change the construction given in Definition 5 in or der to obtain a tight b ound for the minimum distance. W e ca ll this new construction the BQ-Plotkin c onst r u ction . Let A , B and C b e three quater nary linear co des of types ( N ; γ A , δ A ), ( N ; γ B , δ B ), ( N ; γ C , δ C ), with minim um distances d A , d B and d C , respectively . Definition 7 (BQ-Plotkin Construction) L et G A , G B and G C b e gener ator matric es of the quaternary line ar c o des A , B and C , r esp e ctively. We define a new c o de B Q ( A , B , C ) a s t he quaternary l ine ar c o de gener ate d by G B Q =     G A G A G A G A 0 G ′ B 2 G ′ B 3 G ′ B 0 0 ˆ G B ˆ G B 0 0 0 G C     , wher e G ′ B is t he matrix obtaine d fr om G B after switching t wos by ones in their γ B r ows of or der two and ˆ G B is the matrix obtai ne d fr om G B after re moving their γ B r ows of or der two. Prop osition 8 The quaternary line ar c o de B Q ( A , B , C ) is of typ e (4 N ; γ , δ ) , wher e γ = γ A + γ C and δ = δ A + γ B + 2 δ B + δ C ; the binary length is n = 8 N ; the size is 2 γ +2 δ and the minimum distanc e d = min { 4 d A , 2 d B , d C } . 7 Pr o of: The type, the length and the size of B Q ( A , B , C ) can b e easily com- puted fr o m the d efinition of t he co de. T o chec k the minimum distance of B Q ( A , B , C ) let us consider any vector u = ( u 1 | u 1 | u 1 | u 1 ) + ( 0 | u 2 | 2 u 2 | 3 u 2 ) + ( 0 | 0 | u 3 | u 3 ) + ( 0 | 0 | 0 | u 4 ) ∈ B Q ( A , B , C ), wher e u 1 ∈ A ; u 2 ∈ B ′ ; u 3 ∈ ˆ B and u 4 ∈ C . Co des B ′ and ˆ B are the quaternary linear co des genera ted by G ′ B and ˆ G B , respectively . Since B Q ( A , B , C ) is a q uaternary linear code it is enough to show that the weight of u is at lea st d . If u 2 = 0 then by using the same arguments as in P ropo sition 6 we have w L ( u ) ≥ min { 4 d A , 2 d B , d C } b ecause d ˆ B ≥ d B . If u 2 6 = 0 then we distinguish t w o cases. If u 4 = 0 then w L ( u ) = w L ( u 1 | u 1 + u 2 | u 1 + 2 u 2 + u 3 | u 1 + 3 u 2 + u 3 ) = (w L ( u 1 ) + w L ( u 1 + u 2 )) + (w L ( u 1 + 2 u 2 + u 3 ) + w L ( u 1 + 2 u 2 + u 3 + u 2 )) = (w L ( u 1 ) + w L ( u 1 + 2 u 2 + u 3 )) + (w L ( u 1 + u 2 )) + w L ( u 1 + 2 u 2 + u 3 + u 2 )) ≥ w L (2 u 2 + u 3 ) + w L (2 u 2 + u 3 ) (by the triangle inequality) ≥ 2w L (2 u 2 + u 3 ) . Note that 2 u 2 ∈ B and u 3 ∈ ˆ B ⊂ B . If u 3 6 = 2 u 2 then w L (2 u 2 + u 3 ) ≥ d B and w L ( u ) ≥ 2 d B . If 2 u 2 = u 3 then u 2 ∈ ˆ B a nd w L ( u 2 ) ≥ d B . So, w L ( u ) = w L ( u 1 | u 1 + u 2 | u 1 | u 1 + u 2 ) ≥ 2w L ( u 2 ) ≥ 2 d B . Using t wice the triangle ineq ualit y , the case u 4 6 = 0 easily gives w L ( u ) = w L ( u 1 | u 1 + u 2 ) + w L ( u 1 + 2 u 2 + u 3 | u 1 + 3 u 2 + u 3 + u 4 ) ≥ w L ( u 2 ) + w L ( u 2 + u 4 ) ≥ w L ( u 4 ) . Hence, d ≥ min { 4 d A , 2 d B , d C } . But the equa lity holds after the following consideratio ns. T ak ing the sp ecific vector u 1 ∈ A w ith minimum weight d A we obta in w L ( u 1 | u 1 | u 1 | u 1 ) = 4 d A . T ak ing the sp ecific vector u 4 ∈ C with minimum weigh t d C we obtain w L ( 0 | 0 | 0 | u 4 ) = d C . T ak ing the sp ecific v ector u 2 ∈ B with minim um weigh t d B we obtain the following. Note that B ⊂ B ′ and s o we can write the vector u 2 as u 2 = ˆ v + 2 w ′ , where ˆ v ∈ ˆ B and w ′ ∈ B ′ \ ˆ B . T ake the vector u 2 = ˆ v + 2 w ′ ∈ B ′ and, moreov er, the vector ˆ u = 2 ˆ v ∈ ˆ B a nd co mpose the vector (0 | u 2 | 2 u 2 + ˆ u | 3 u 2 + ˆ u ) = (0 | u 2 | 0 | u 2 ) which belo ngs to B Q ( A , B , C ). This v ector ha s minim um Lee w eig h t 2 d B .  3 Quaternary Reed-Muller co des The usual binary linear R M fa mily of co des is o ne of the oldest and inter- esting family of co des. The codes in this family are easy to deco de and their combinatorial prop e rties are of great in terest t o produce new optimal co des. 8 F or an y integer m ≥ 1 the family of bina ry linear RM co des is giv en by the sequence R M ( r , m ), where 0 ≤ r ≤ m . The co de RM ( r , m ) is called the r - th order bina r y linear Reed-Muller co de of length n = 2 m and it is tr ue that RM (0 , m ) ⊂ RM (1 , m ) ⊂ . . . ⊂ RM ( r − 2 , m ) ⊂ RM ( r − 1 , m ) ⊂ R M ( r , m ) . Let 0 ≤ r ≤ m , m ≥ 1. F ollowing [1 2] the RM ( r , m ) code of order r can be constructed b y using the Plotkin construction in the following way: RM (0 , m ) = { 0 , 1 } , RM ( m, m ) = Z 2 m 2 , RM ( r , m ) = { ( u 1 | u 1 + u 2 ) : u 1 ∈ R M ( r, m − 1) , u 2 ∈ RM ( r − 1 , m − 1) } . (3) It is imp ortant to no te that if we fix m , once we know the se q uence RM ( r, m ) for all 0 ≤ r ≤ m , then it is easy to obta in the new sequence R M ( r, m + 1) b y using the Plotkin co nstruction (3). Moreov er , the co des in the RM family fulfill the basic prop erties s umma r ized in t he following theorem: Theorem 9 ([12]) The binary line ar R e e d-Mul ler family of c o des { R M ( r , m ) } , 0 ≤ r ≤ m , has the fol lowing pr op ert ies: 1. the length n = 2 m , m ≥ 1 ; 2. the minimum distanc e d = 2 m − r ; 3. the dimension k = r X i =0  m i  ; 4. the c o de R M ( r − 1 , m ) is a s ub c o de of RM ( r, m ) , r > 0 . T he c o de RM (0 , m ) is the r ep etition c o de with only one nonzer o c o dewor d (the al l ones ve ctor). The c o de R M ( m, m ) is t he whole sp ac e Z 2 m 2 and R M ( m − 1 , m ) is the even c o de (that is, the c o de with al l the ve ctors of even wei ght fr om Z 2 m 2 ); 5. the c o de RM (1 , m ) is the binary line ar Hadamar d c o de and RM ( m − 2 , m ) is the extende d binary H amming c o de of p ar ameters (2 m , 2 m − m − 1 , 4) ; 6. the c o de RM ( r, m ) is the dual c o de of R M ( m − 1 − r, m ) for 0 ≤ r < m . In the recent litera ture [7, 20, 2, 3 ] several families of quaternary linear co des hav e been pr o posed and studied trying to genera lize the RM co des. How ever, when we take the c orresp onding Z 4 -linear co des, they do not sa tisfy all the prop erties in Theorem 9. This last requiremen t is the main go a l of the pre s en t work, to construct new families of quaternary linear co des such that, after the Gray ma p, we obtain Z 4 -linear co des with the parameters and prop erties quo ted in Th eo rem 9. The result of the present pap er gene r alizes the results in [19]. F urther w e will refer to these quaternar y linear Reed-Muller co des as RM to distinguish them from the binary linear Reed-Muller co des R M . Co n trar y to the binary linear case, where there is only one RM family , in the quater nary case we hav e ⌊ m +1 2 ⌋ families for each v alue of m . W e will distinguish these families by us ing subindexes s from t he set { 0 , . . . , ⌊ m − 1 2 ⌋} . 9 T able 1: RM ( r, m ) codes f or m = 1 ( r , m ) (0,1) (1,1) N ( γ , δ ) 1 ( 1,0 ) (0,1) RM 0 ( r , 1) 3.1 The family of RM ( r, 1) co des W e beg in b y considering the trivial case of m = 1 , that is, the case of co des of binary leng th n = 2 1 . The q uaternary linear Reed-Mulle r co de RM (0 , 1) is the rep etition co de with only one nonzero co dew or d (the vector with only one qua ternary co ordina te of v alue 2). This quater nary linear code is of t yp e (1; 1 , 0). The co de RM (1 , 1) is the whole space Z 1 4 , so a quaterna ry linear co de of type (1; 0 , 1). These tw o co des, RM (0 , 1) and RM (1 , 1), after the Gra y map, give binary co des with the same par ameters o f the corres ponding binary co des RM ( r, 1 ) a nd with the same prop erties describ ed in Theorem 9. In this case, when m = 1, not only these codes hav e the same par ameters, but they have the same codewords. W e will r efer to thes e co des as RM 0 (0 , 1 ) and RM 0 (1 , 1 ), resp ectively , as it is sho wn in T able 1. In eac h en try of th is table there are t he parameter s ( γ , δ ) of the corresp onding co de of t yp e ( N ; γ , δ ). Since we will need a n s pecific repr esen tation for these co des in T able 1, w e will agree in using further the follo wing matrices as the generator matrices for each o ne of them . The generato r matr ix of R M 0 (0 , 1 ) is G 0 (0 , 1 ) =  2  and the g enerator matrix of RM 0 (1 , 1 ) is G 0 (1 , 1 ) =  1  . 3.2 Plotkin and BQ-Plotkin const ructions The first imp ortant po in t is to apply the P lotkin construction to quaternar y linear Reed-Muller co des. Let RM s ( r , m − 1) and RM s ( r − 1 , m − 1), 0 ≤ s ≤ ⌊ m − 2 2 ⌋ , b e any tw o RM co des o f type ( N ; γ s r,m − 1 , δ s r,m − 1 ) and ( N ; γ s r − 1 ,m − 1 , δ s r − 1 ,m − 1 ); binar y length n = 2 m − 1 ; num b er of co dewords 2 k r and 2 k r − 1 ; minimum distance 2 m − r − 1 and 2 m − r resp ectiv ely , where k r = r X i =0  m − 1 i  , k r − 1 = r − 1 X i =0  m − 1 i  . Theorem 10 F or a ny r and m ≥ 2 , 0 < r < m , the c o de obtaine d by u sing the Plotkin c onstru ction RM s ( r , m ) = { ( u 1 | u 1 + u 2 ) : u 1 ∈ RM s ( r , m − 1) , u 2 ∈ RM s ( r − 1 , m − 1) } , 10 T able 2: RM ( r, m ) codes f or m = 2 ( r , m ) (0 , 2 ) (1 , 2) (2 , 2) N ( γ , δ ) 2 ( 1 , 0) (1 , 1) (0 , 2) RM 0 ( r , 2) wher e 0 ≤ s ≤ ⌊ m − 1 2 ⌋ , is a quaternary line ar c o de of t yp e ( 2 N ; γ s r,m , δ s r,m ), wher e γ s r,m = γ s r,m − 1 + γ s r − 1 ,m − 1 and δ s r,m = δ s r,m − 1 + δ s r − 1 ,m − 1 ; the binary length is n = 2 m ; the numb er of c o dewor ds is 2 k , wher e k = r X i =0  m i  , the c o de distanc e is 2 m − r and RM s ( r − 1 , m ) ⊂ RM s ( r , m ) . F or r = 0 , t he c o de RM s (0 , m ) is the r ep etition c o de wi th only one nonzer o c o dewor d (the al l twos ve ctor). F or r = m , t he c o de RM s ( m, m ) is the whole sp ac e Z 2 m − 1 4 . Pr o of: The t yp e (2 N ; γ s r,m , δ s r,m ) of the co de RM s ( r , m ), its size and the minim um distance ca n be computed fr om Prop osition 2. Since RM s ( r − 1 , m − 1) ⊂ RM s ( r , m − 1) and RM s ( r − 2 , m − 1) ⊂ RM s ( r − 1 , m − 1), then taking into account the co des g iv en in the p r evious section b y inductio n we g et RM s ( r − 1 , m ) ⊂ RM s ( r , m ).  F or m = 2, taking the RM 0 ( r , 1) co des in T able 1 a nd applying Theore m 10 we obtain the codes in T able 2 . The g enerator matrice s for these codes are the following RM 0 (0 , 2 ) :  2 2  ; RM 0 (1 , 2 ) :  0 2 1 1  ; RM 0 (2 , 2 ) :  1 0 0 1  . (4) F or m = 3 , it is w ell kno wn that there exist t wo Z 4 -linear Hadamard c odes [10]. So, our g oal is to co nstruct tw o families of quaternary linea r Reed- Muller co des a s it is shown in T a ble 3. The co des in the first r o w of T able 3 can be obtained due to the P lotkin construction from the co des of T able 2. But, the co des in the second row ca n not b e obta ined by using only the Plo tkin construction. It is in this ca se that we need to exploit the new BQ-P lotkin construction a s w e will see later in this section. Constructions o f additive c o des with the par ameters of the binary linea r Reed-Muller co des by using only the Plotkin construction were initiated in [1 6, 19]. Lemma 11 L et {A i } , i = 1 , 2 , 3 , 4 , b e a family of four quaternary line ar c o des of typ es ( N ; γ i , δ i ) with gener ator matric es G i , r esp e ct ively . L et A ′ i and ˆ A i b e 11 T able 3: RM ( r, m ) codes f or m = 3 ( r , m ) (0 , 3 ) (1 , 3) (2 , 3) (3 , 3) N ( γ , δ ) 4 (1 , 0 ) (2 , 1) (1 , 3) (0 , 4) RM 0 ( r , 3) 4 (1 , 0 ) (0 , 2) (1 , 3) (0 , 4) RM 1 ( r , 3) the c o des gener ate d by G ′ i and ˆ G i , r esp e ctively, such that for i = 1 , 2 , 3 it is true that (i) A i ⊂ A i +1 ; (ii) ˆ A i ⊂ ˆ A i +1 ; (iii) A ′ i ⊂ A ′ i +1 ; (iv) A ′ i ⊂ A i +1 . Then, the family {P C ( A i +1 , A i ) } of the thr e e c o des P C ( A 2 , A 1 ) , P C ( A 3 , A 2 ) and P C ( A 4 , A 3 ) satisfies (i), (ii), ( iii) and (iv) for i = 1 , 2 and t he family {B Q ( A i +2 , A i +1 , A i ) } of the t wo c o des B Q ( A 3 , A 2 , A 1 ) and B Q ( A 4 , A 3 , A 2 ) satisfies the pr op erties (i), (ii), (iii) and (iv) for i = 1 . Pr o of: It is str a igh tforward to see that the P lo tkin construction fulfills the prop erties. F or the BQ-Plo tkin co nstruction the prop erty (i) is clear from Definition 7. Now, the ge ne r ator matrix of ˆ B Q ( A i +2 , A i +1 , A i ) ha s the follo wing f o r m:     ˆ G i +2 ˆ G i +2 ˆ G i +2 ˆ G i +2 0 G ′ i +1 2 G ′ i +1 3 G ′ i +1 0 0 ˆ G i +1 ˆ G i +1 0 0 0 ˆ G i     . (5) Using the prop erties (ii) and (iii) for the matrices G i , G i +1 , G i +2 , we get the pr oper t y (ii) for B Q ( A i +2 , A i +1 , A i ) a nd i = 1. Since, th e generator matrix of B Q ′ ( A i +2 , A i +1 , A i ) ha s the form     G ′ i +2 G ′ i +2 G ′ i +2 G ′ i +2 0 G ′ i +1 2 G ′ i +1 3 G ′ i +1 0 0 ˆ G i +1 ˆ G i +1 0 0 0 G ′ i     , (6) using the prop erties (ii) and (iii) for the matrices G i , G i +1 , G i +2 we obtain the prop erty (iii) for B Q ( A i +2 , A i +1 , A i ) a nd i = 1 . 12 Finally , the g enerator matrix of B Q ( A i +2 , A i +1 , A i ) ha s the form     G i +2 G i +2 G i +2 G i +2 0 G ′ i +1 2 G ′ i +1 3 G ′ i +1 0 0 ˆ G i +1 ˆ G i +1 0 0 0 G i     . (7) Using the pr oper ties (ii) , (iii) and (iv) for the matr ices G i , G i +1 , G i +2 we obtain the pr oper t y (iv) for B Q ( A i +2 , A i +1 , A i ) a nd i = 1 .  Let RM s − 1 ( r , m − 2), RM s − 1 ( r − 1 , m − 2) and RM s − 1 ( r − 2 , m − 2), 0 < s ≤ ⌊ m − 3 2 ⌋ , m > 3, be any three RM co des of type ( N ; γ s − 1 r,m − 2 , δ s − 1 r,m − 2 ), ( N ; γ s − 1 r − 1 ,m − 2 , δ s − 1 r − 1 ,m − 2 ) and ( N ; γ s − 1 r − 2 ,m − 2 , δ s − 1 r − 2 ,m − 2 ); binary length n = 2 m − 2 ; nu mber of co dew ords 2 k r , 2 k r − 1 and 2 k r − 2 ; minim um dista nces 2 m − r − 2 , 2 m − r − 1 and 2 m − r resp ectiv ely , where k r = r X i =0  m − 2 i  , k r − 1 = r − 1 X i =0  m − 2 i  , k r − 2 = r − 2 X i =0  m − 2 i  . Let G s ( r , m ), 0 < r < m − 1, b e the matrix     G s − 1 ( r , m − 2) G s − 1 ( r , m − 2) G s − 1 ( r , m − 2) G s − 1 ( r , m − 2) 0 G ′ s − 1 ( r − 1 , m − 2) 2 G ′ s − 1 ( r − 1 , m − 2 ) 3 G ′ s − 1 ( r − 1 , m − 2) 0 0 ˆ G s − 1 ( r − 1 , m − 2) ˆ G s − 1 ( r − 1 , m − 2) 0 0 0 G s − 1 ( r − 2 , m − 2)     (8) F or the special cas e r = 1 we need to define G s − 1 ( − 1 , m − 2) as the g enerator matrix o f the a ll zer o codeword co de. Theorem 12 F or any r and m ≥ 3 , 0 < r < m − 1 , the RM s ( r , m ) c o de, 0 < s ≤ ⌊ m − 1 2 ⌋ , obtaine d by using the BQ-Plotkin c onst ruction in Defin it ion 7 and with the gener ator matrix G s ( r , m ) define d in (8), is a quaternary line ar c o de of typ e ( 4 N ; γ s r,m , δ s r,m ), wher e γ s r,m = γ s − 1 r,m − 2 + γ s − 1 r − 2 ,m − 2 ; δ s r,m = δ s − 1 r,m − 2 + γ s − 1 r − 1 ,m − 2 + 2 δ s − 1 r − 1 ,m − 2 + δ s − 1 r − 2 ,m − 2 ; the binary length is n = 2 m ; the numb er of c o dewor ds is 2 k , wher e k = r X i =0  m i  ; the minimum distanc e is 2 m − r and RM s ( r − 1 , m ) ⊂ RM s ( r , m ) . Pr o of: T he t yp e (4 N ; γ s r,m , δ s r,m ) of the code RM s ( r , m ) and the minim um distance ca n b e computed from Prop osition 8. T o compute the size note that 2 k = |RM s ( r , m ) | = |RM s − 1 ( r , m − 2) | × |RM ′ s − 1 ( r − 1 , m − 2) | ×| ˆ RM s − 1 ( r − 1 , m − 2) | × |RM s − 1 ( r − 2 , m − 2 ) | , 13 where RM ′ s − 1 ( r − 1 , m − 2 ) a nd ˆ RM s − 1 ( r − 1 , m − 2 ) a re the quaternary linear co des generated by G ′ s − 1 ( r − 1 , m − 2) and ˆ G s − 1 ( r − 1 , m − 2), resp ectively . Hence, |RM ′ s − 1 ( r − 1 , m − 2) | × | ˆ RM s − 1 ( r − 1 , m − 2) | = 2 2 γ s − 1 r − 1 ,m − 2 +4 δ s − 1 r − 1 ,m − 2 = 2 2 k r − 1 So, k = k r + 2 k r − 1 + k r − 2 . Finally , w e obtain k = r X i =0  m − 2 i  + 2 r − 1 X i =0  m − 2 i  + r − 2 X i =0  m − 2 i  = r X i =0  m − 1 i  + r − 1 X i =0  m − 1 i  = r X i =0  m i  . T o prov e that RM s ( r − 1 , m ) ⊂ RM s ( r , m ) notice that fr o m Lemma 11 and since the codes of T able 1 and T able 2 fulfill the four conditions of this lemma we c an conclude b y induction that the co de gene r ated b y the ma trix G s ( r − 1 , m ) is a subco de of the code generated by the matrix G s ( r , m ).  F or every 0 < s ≤ ⌊ m − 1 2 ⌋ th e family of co des RM s ( r , m ) constructed using the ab ov e theorem is incomplete in the sens e that the co des RM s ( − 1 , m ), RM s (0 , m ), RM s ( m − 1 , m ), RM s ( m, m ) do not co me fro m the construction. T o b e coherent with all the notations, for r = − 1, the c ode RM s ( − 1 , m ) is defined as the a ll zero co deword co de. F o r r = 0, the co de RM s (0 , m ) is defined as the rep etition co de with o nly one no n zero co dew or d (the all t wos quater nary vector). F or r = m − 1 and r = m , t he co des RM s ( m − 1 , m ) and RM s ( m, m ) are defined as the even weigh t co de a nd the whole spa ce Z 2 m − 1 4 , resp ectively . The constr uction of the families of Reed-Muller c o des in Theorem 1 2 is based on the generator matrices and so, for ea c h index s , we need a generator matrix for the co des RM s ( − 1 , m ), RM s (0 , m ), RM s ( m − 1 , m ), RM s ( m, m ). W e will use the following genera tor matrices : G s ( − 1 , m ) =  0 · · · 0  , G s (0 , m ) =  2 · · · 2  , G s ( m, m ) = I 2 m − 1 . The ge nerator matrix G s ( m − 1 , m ) will b e re cursively obtained by using the BQ-P lotkin construction B Q ( RM s − 1 ( m − 2 , m − 2) , RM s − 1 ( m − 2 , m − 2) , RM s − 1 ( m − 3 , m − 2)) (see Definition7). Prop osition 13 F or m ≥ 3 , the matrix G s ( m − 1 , m ) of Definition 7 asso ciate d to B Q ( RM s − 1 ( m − 2 , m − 2) , RM s − 1 ( m − 2 , m − 2) , RM s − 1 ( m − 3 , m − 2)) is a gener ator matrix of RM s ( m − 1 , m ) . Pr o of: All the rows in matr ix G s ( m − 1 , m ) a re vectors of ev en weigh t. So, to prov e that this matr ix gener ates RM s ( m − 1 , m ) w e only need to chec k if the dimens io n is the adequate. W e will prove, b y induction on m ≥ 1 , that γ s m − 1 ,m = 1 and δ s m − 1 ,m = 2 m − 1 − 1. The claim is triv ia lly true fo r m = 1 and m = 2 us ing the matrices defined in section 3 .1 and in (4). Supp ose t he claim is true fo r m ≥ 2. 14 T able 4: RM ( r, m ) codes f or m = 4 ( r , m ) (0 , 4 ) (1 , 4) (2 , 4) (3 , 4) (4 , 4) N ( γ , δ ) 8 (1 , 0 ) (3 , 1) (3 , 4) (1 , 7) (0 , 8) RM 0 ( r , 4) 8 (1 , 0 ) (1 , 2) (1 , 5) (1 , 7) (0 , 8) RM 1 ( r , 4) Since the matrix G s ( m − 1 , m ) is upp er triangula r, we must to a dd the resp ectiv e v alues γ a nd δ of G s − 1 ( m − 2 , m − 2 ), G ′ s − 1 ( m − 2 , m − 2), ˆ G s − 1 ( m − 2 , m − 2) and G s − 1 ( m − 3 , m − 2 ). By induction, γ s − 1 m − 3 ,m − 2 = 1 and δ s − 1 m − 3 ,m − 2 = 2 m − 3 − 1. Since G ′ s − 1 ( m − 2 , m − 2) = ˆ G s − 1 ( m − 2 , m − 2) = G s − 1 ( m − 2 , m − 2 ) w e have γ s − 1 m − 2 ,m − 2 = 0 and δ s − 1 m − 2 ,m − 2 = 2 m − 3 . Hence, we obtain γ s m − 1 ,m = 0 + 0 + 0 + 1 = 1, δ s m − 1 ,m = 2 m − 3 + 2 m − 3 + 2 m − 3 + 2 m − 3 − 1 = 2 m − 1 − 1.  Note that with these definitions o f RM s ( − 1 , m ), RM s (0 , m ), RM s ( m − 1 , m ), RM s ( m, m ), the family of co des RM s ( r , m ), 0 ≤ r ≤ m , 0 < s ≤ ⌊ m − 1 2 ⌋ , fulfills the four co nditions of Lemma 11. Using Theorems 10 and 12 we ca n construct the RM co de s in the t wo rows of T able 3. W e do no t write the generato r matrices for co de s RM 0 ( r , 3) beca use they can b e directly obtained fro m the resp ective codes for m = 2 b y using the Plotkin construction. F or the co des in the family RM 1 ( r , 3) we pr esen t the generator matr ices as a direct application of Theorem 12: RM 1 (0 , 3 ) :  2 2 2 2  ; RM 1 (1 , 3 ) :  1 1 1 1 0 1 2 3  ; RM 1 (2 , 3 ) :     0 0 0 2 1 1 1 1 0 1 2 3 0 0 1 1     (9) and the remaining co de R M 1 (3 , 3 ) in t he family is the whole space Z 2 2 4 . All these co des, after the Gray map, give binar y co des with the same parame- ters as the co des RM ( r, 3 ) and with the s ame prop erties desc r ibed in Theor em 9. In the case m = 3 , like in the case m = 2 not only these co des hav e the sa me parameters , but they hav e the same co dew ords . T his is not in this w ay for all the o ther v alues m > 3. Now, fro m T able 3 and b y using the Plo tkin construction we c an construct the t wo f amilies of the co des RM s ( r , 4) for s = 0 , 1, as it is sho wn in T able 4. Note that the family o f co des RM 1 ( r , 4) also can b e obtained using the BQ - Plotkin construction from th e family of codes RM 0 ( r , 2) in T a ble 2. F ro m the co des in T able 4 applying the P lotkin construction w e can construct the tw o families of RM s ( r , 5), s = 0 , 1 , as it is shown in T able 5. The thir d 15 T able 5: RM ( r, m ) codes f or m = 5 ( r , m ) (0 , 5 ) (1 , 5) (2 , 5) (3 , 5 ) (4 , 5) (5 , 5) N ( γ , δ ) 16 (1 , 0) (4 , 1) (6 , 5) (4 , 11) (1 , 15) (0 , 16) RM 0 ( r , 5) 16 (1 , 0) (2 , 2) (2 , 7) (2 , 12) (1 , 15) (0 , 16) RM 1 ( r , 5) 16 (1 , 0 ) (0 , 3) (2 , 7) (0 , 13) (1 , 15) (0 , 16) RM 2 ( r , 5) family in T able 5, RM 2 ( r , 5), is obtained applying the BQ-Plotkin construction to the RM 1 ( r , 3) family of T able 3 . Note tha t RM 0 ( r , 5) only can be obtained applying the Plo tkin co nstruction, RM 2 ( r , 5) only can b e o bta ined applying the BQ-Plo tkin constr uction, but RM 1 ( r , 5) can be obtained by using the Plo tkin or the BQ -Plotkin constr uction. In gener al, for m > 1, the co de RM 0 ( r , m ) can b e only o btained applying the Plotkin constructio n. F or m even a nd m o dd, but s 6 = m − 1 2 , families of RM s ( r , m ) can be obta ined a pplying the P lotkin or the BQ-Plo tkin constr uc- tion. F o r m odd and s = m − 1 2 , RM s ( r , m ) only can be obtained applying the BQ-Plo tkin constr uction. A question ar ises at this point, how ma ny families o f Reed-Muller codes can b e obta ined combining the Plotkin and the BQ- Plotkin constructions? Next pr opositio n prov es that no new co des app ear when we combine both these constructions. Given three q uaternary line a r codes A , B and C , we remind t hat P C ( A , B ) is the quater nary linear co de obtained applying the Plotkin cons tr uction (see Definition 1) and B Q ( A , B , C ) is the q uaternary linear co de obtained by using the BQ-P lotkin construction (see Definition 7). The following pro position shows that t he t wo constructions comm ute. Prop osition 14 Given four quaternary line ar c o des A , B , C and D , then the c o des P C ( B Q ( A , B , C )) , B Q ( B , C , D )) and BQ ( P C ( A , B ) , P C ( B , C ) , P C ( C , D )) ar e p er- mutational ly e quivalent. The proof is straightforw ar d. Notice that the sa me result is true changing the B Q-Plotkin construction b y the q uaternary Plo tkin construction or the double Plotkin construction. F ro m now on, when w e talk abo ut the family o f Reed-Muller co des {RM s ( r , m ) } constructed b y using the Plotkin and the BQ-Plo tk in constructions we will as- sume that for m even and m o dd, but s 6 = m − 1 2 , these families of co des are obtained a pplying the Plotkin co nstruction. F or m o dd and s = m − 1 2 , the family of co des is obtained applying the BQ-P lotkin construction. The following lemma computes the v a lues for the parameters γ and δ of the RM s ( r , m ) c odes in the specific case when m is odd, m ≥ 3 a nd s = m − 1 2 . 16 Lemma 15 F or o dd m , m ≥ 3 and s = m − 1 2 we have the fol lowing values for the p ar ameters γ s r,m and δ s r,m of the RM s ( r , m ) c o de built by using the BQ- Plotkin c onstru ction with the g ener ator matrix (8): (i) F or o dd r it is true that γ s r,m = 0 . (ii) F or even r we have γ s r,m =  ( m − 1) / 2 r / 2  . (iii) The fol lowing e qualities δ s m,m = 2 m − 1 , δ s m − 1 ,m = 2 m − 1 − 1 and δ s m − 2 ,m = 2 m − 1 − m +1 2 ar e tr u e. Pr o of: Note that b y Propo s ition 8 it is true that γ s r,m = γ s − 1 r,m − 2 + γ s − 1 r − 2 ,m − 2 with γ 0 0 , 1 = 1 and γ 0 1 , 1 = 0. Using induction w e ca n prov e (i) a nd (ii) . Clearly , δ s m,m = 2 m − 1 , δ s m − 1 ,m = 2 m − 1 − 1 . The v alue of m is o dd, hence m − 2 is also o dd and γ s m − 2 ,m = 0 . So, |RM s ( m − 2 , m ) | = 2 2 δ s m − 2 ,m but, also, |RM s ( m − 2 , m ) | = 2 2 m − ( m m − 1 ) − ( m m ) . Finally , 2 δ s m − 2 ,m = 2 m − m − 1 and δ s m − 2 ,m = 2 m − 1 − m +1 2 .  As it is prov ed in Theorems 1 0 and 1 2 the constructed families of RM co des satisfy the same pro p erties we stated fo r bina r y linear Reed-Muller codes in Theor em 9 except for the duality . In the following Sectio n we will discuss this t opic . Notice that, a fter the Gray map, the c onstructed RM fa milies of qua ter - nary linear Reed-Muller co de s hav e no t only the sa me par ameters as the usual binary linear family of R M co des, but a lso the characteristic prop erties of co des RM s (1 , m ) and RM s ( m − 2 , m ) as it is stated in the follo wing pro position. Prop osition 16 F or any inte ger m ≥ 1 and 0 ≤ s ≤ ⌊ m − 1 2 ⌋ , after the Gr ay map t he c o de RM s (1 , m ) is a Z 4 -line ar H adamar d c o de and the c o de RM s ( m − 2 , m ) i s an Z 4 -line ar extende d p erfe ct c o de. Pr o of: F rom Theore m 12 we hav e that the codes RM s (1 , m ), where 0 ≤ s ≤ ⌊ m − 1 2 ⌋ , are quaternary linear and, under the Gr a y map, ha ve the par a meters of Hadamard code s . Ana lo gously all the codes RM s ( m − 2 , m ), 0 ≤ s ≤ ⌊ m − 1 2 ⌋ , after the Gra y map, are Z 4 -linear a nd ha ve the par ameters of ex tended p erfect binary co des. By the Krotov classification [9, 10], these co des co uld b e o nly Z 4 -linear Hadamard and extended Z 4 -linear perfect codes, respectively .  4 Dualit y F or the usual binary linear RM co des w e know that the co des R M ( r , m ) and RM ( m − r − 1 , m ) ar e dual to eac h o ther. The fa milies of RM co des hav e the same pr oper ty if we use the K roneck er inner pr oduct to define the Z 4 -duality . 17 Throughout this section the notion of dualit y will be r elated to the Krone cker inner pr o duct defined in (2). W e b egin by studying the duality prop erties for the family of RM co des obtained by using the BQ-Plo tkin construction, that is, we are going to prove the dualit y relationships for the family o f co des RM s ( r , m ), m odd, s = m − 1 2 , constructed f ro m the f amily RM s − 1 ( r , m − 2). Basically , we will prove this fact by induction but, previously , w e need tw o techn ica l lemmas. W e will use G s ( r , m ) to refer to t he g enerator matrix of co de RM s ( r , m ); the matrices G ′ s ( r , m ), ˆ G s ( m − r − 1 , m ) will hav e the meaning that we in tro duced in Definition 7 and RM ′ s ( r , m ), ˆ RM s ( m − r − 1 , m ) will be the co des genera ted b y G ′ s ( r , m ) a nd ˆ G s ( m − r − 1 , m ), respectively . Lemma 17 L et u , v ∈ Z N 4 b e any two ve ctors such t hat u = ( u 1 | u 2 ) and v = ( v 1 | v 2 ) , wher e u 1 , u 2 , v 1 , v 2 ∈ Z N/ 2 4 . Then, h ( u 1 | u 2 ) , ( v 1 | v 2 ) i ⊗ N = h u 1 , v 1 i ⊗ N/ 2 + 3 h u 2 , v 2 i ⊗ N/ 2 . (10) Pr o of: Straig h tforward from the Kroneck er inner product definition.  Lemma 18 L et m b e an o dd inte ger, m ≥ 3 , N = 2 m − 1 and s = m − 1 2 . L et { RM s ( r , m ) } b e the family of RM c o des obtaine d in The or em 12 by us- ing t he BQ-Plotkin c onstruction. Then, for e ach 0 ≤ r ≤ m , for al l u ∈ G ′ s ( r , m ) \ ˆ G s ( r , m ) and v ∈ ˆ RM s ( m − r − 1 , m ) we have h u , v i ⊗ N = 0 . Pr o of: W e pro ceed b y induction on m be g inning with m = 3. Using (9) it is easy to see that the as sertion is true for m = 3. F or the case whe n r is o dd the statement is trivially true, since there is nothing to pro of. Indeed, from Lemma 15, γ s r,m = 0 and so G ′ s ( r , m ) \ ˆ G s ( r , m ) = ∅ . Hence, along this proof w e can take r as a n even in teger . Now, for m > 3 and 0 < r ≤ m − 2 , assume b y induction hypothesis that for a ll x ∈ G ′ s − 1 ( r , m − 2) \ ˆ G s − 1 ( r , m − 2) and y ∈ ˆ RM s ( m − r − 3 , m − 2) is h x , y i ⊗ N/ 4 = 0. Let v ∈ ˆ RM s ( m − r − 1 , m ) and u ∈ G ′ s ( r , m ) \ ˆ G s ( r , m ), 0 < r ≤ m − 2 . W e will pro ve by inductio n that h u , v i ⊗ N = 0. F ro m P ropo sition 8, we hav e u = ( u 1 | u 1 | u 1 | u 1 ) + ( 0 | 0 | 0 | u 4 ), wher e u 1 ∈ G ′ s − 1 ( r , m − 2) \ ˆ G s − 1 ( r , m − 2) and u 4 ∈ G ′ s − 1 ( r − 2 , m − 2) \ ˆ G s − 1 ( r − 2 , m − 2). Also, we have v = ( v 1 | v 1 | v 1 | v 1 ) + ( 0 | v 2 | 2 v 2 | 3 v 2 ) + ( 0 | 0 | v 3 | v 3 ) + ( 0 | 0 | 0 | v 4 ) with v 1 ∈ ˆ RM s − 1 ( m − r − 1 , m − 2 ), v 2 ∈ RM ′ s − 1 ( m − r − 2 , m − 2 ), v 3 ∈ ˆ RM s − 1 ( m − r − 2 , m − 2) and v 4 ∈ ˆ RM s − 1 ( m − r − 3 , m − 2). Now, b y using Lemma 17: h u , v i ⊗ N = 8 h u 1 , v 1 i ⊗ N/ 4 + 12 h u 1 , v 2 i ⊗ N/ 4 + 4 h u 1 , v 3 i ⊗ N/ 4 + h u 1 , v 4 i ⊗ N/ 4 + h u 4 , v 1 i ⊗ N/ 4 + 3 h u 4 , v 2 i ⊗ N/ 4 + h u 4 , v 3 i ⊗ N/ 4 + h u 4 , v 4 i ⊗ N/ 4 = h u 1 , v 4 i ⊗ N/ 4 + h u 4 , v 1 i ⊗ N/ 4 + h u 4 , z i ⊗ N/ 4 , where z = 3 v 2 + v 3 + v 4 . 18 By induction hypo thesis h u 1 , v 4 i ⊗ N/ 4 = h u 4 , v 1 i ⊗ N/ 4 = 0 and so we need only to sho w that h u 4 , z i ⊗ N/ 4 = 0. F ro m Lemma 11 we hav e ˆ RM s − 1 ( m − r − 3 , m − 2) ⊂ ˆ RM s − 1 ( m − r − 2 , m − 2) ⊂ ˆ RM s − 1 ( m − r − 1 , m − 2) and so v 3 + v 4 ∈ ˆ RM s − 1 ( m − r − 1 , m − 2). As we said at the b eginning of the pr oo f, r is even. Therefor e, w e have that m − r − 2 is o dd and from Lemma 15, we obtain γ s − 1 m − r − 2 ,m − 2 = 0. Hence, G ′ s − 1 ( m − r − 2 , m − 2 ) = ˆ G s − 1 ( m − r − 2 , m − 2) and z = 3 v 2 + v 3 + v 4 ∈ ˆ RM s − 1 ( m − r − 1 , m − 2). But u 4 ∈ G ′ s − 1 ( r − 2 , m − 2) \ ˆ G s − 1 ( r − 2 , m − 2). Then, b y induction h yp othesis, h u 4 , z i ⊗ N/ 4 = 0. Finally , we prov e the statemen t for r = 0; r = m − 1 and r = m. F or r = 0, we pro ceed b y induction. Case m = 3 is trivially true taking int o a ccoun t (9). W e hav e u ∈ G ′ s (0 , m ) \ ˆ G s (0 , m ) and so u is the all ones vector u = (1 , 1 , . . . , 1). An y vector v ∈ ˆ RM s ( m − 1 , m ) is generated b y the rows of ˆ G s ( m − 1 , m ), where G s ( m − 1 , m ) is the ma trix defined in (8). Hence, h u , v i ⊗ N = h u 1 , v 4 i ⊗ N/ 4 = 0, by induction hypothesis, since u 1 ∈ G ′ s − 1 (0 , m − 2) \ ˆ G s − 1 (0 , m − 2) and v 4 ∈ ˆ RM s − 1 ( m − 3 , m − 2). F or r = m − 1 we hav e u = (0 , 0 , . . . , 0 , 1) ∈ G ′ s ( m − 1 , m ) \ ˆ G s ( m − 1 , m ) and v = (0 , 0 , . . . , 0) ∈ ˆ G s (0 , m ), therefore h u , v i ⊗ N = 0. Finally , for r = m the cla im is trivially true, be c ause the set G ′ s ( m, m ) \ ˆ G s ( m, m ) is empty .  Theorem 19 L et m b e an o dd inte ger, m ≥ 1 , N = 2 m − 1 and s = m − 1 2 the set {RM s ( r , m ) } b e the family of RM c o des obtaine d in The or em 12 by u sing the BQ-Plotkin c onstru ction. Then, for e ach 0 ≤ r ≤ m , the c o de RM s ( r , m ) is a quaternary dual of the c o de RM s ( m − r − 1 , m ) . Pr o of: Since |RM s ( r , m ) | · |RM s ( m − r − 1 , m ) | = 2 m , it suffices to pr o ve that fo r every u ∈ G s ( r , m ) and v ∈ G s ( m − r − 1 , m ) we have h u , v i ⊗ N = 0. W e pro ceed by induction o n m . The claim is trivially true for m = 1. Now, for m > 1 and 0 ≤ r ≤ m , assume by induction hypo thesis that for a ll x ∈ G s − 1 ( r , m − 2) and y ∈ G s − 1 ( m − r − 3 , m − 2) it is true that h x , y i ⊗ N/ 4 = 0. Let u ∈ G s ( r , m ) and v ∈ G s ( m − r − 1 , m ) for any 0 < r ≤ m . When 0 < r ≤ m − 2 w e ca n use the follo wing expressions for u and v : u = ( u 1 | u 1 u 1 | u 1 ) + ( 0 | u 2 | 2 u 2 | 3 u 2 ) + ( 0 | 0 | u 3 | u 3 ) + ( 0 | 0 | 0 | u 4 ) , wher e u 1 ∈ G s − 1 ( r , m − 2), u 2 ∈ G ′ s − 1 ( r − 1 , m − 2) u 3 ∈ ˆ G s − 1 ( r − 1 , m − 2) and u 4 ∈ G s − 1 ( r − 2 , m − 2); v = ( v 1 | v 1 | v 1 | v 1 ) + ( 0 | v 2 | 2 v 2 | 3 v 2 ) + ( 0 | 0 | v 3 | v 3 ) + ( 0 | 0 | 0 | v 4 ) , wher e v 1 ∈ G s − 1 ( m − r − 1 , m − 2), v 2 ∈ G ′ s − 1 ( m − r − 2 , m − 2), v 3 ∈ ˆ G s − 1 ( m − r − 2 , m − 2) and v 4 ∈ G s − 1 ( m − r − 3 , m − 2). 19 Therefore, applying Lemma 17 we get h u , v i ⊗ N = 8 h u 1 , v 1 i ⊗ N/ 4 + 12 h u 1 , v 2 i ⊗ N/ 4 + 4 h u 1 , v 3 i ⊗ N/ 4 + h u 1 , v 4 i ⊗ N/ 4 + 12 h u 2 , v 1 i ⊗ N/ 4 + 24 h u 2 , v 2 i ⊗ N/ 4 + 9 h u 2 , v 3 i ⊗ N/ 4 + h u 2 , v 4 i ⊗ N/ 4 + 4 h u 3 , v 1 i ⊗ N/ 4 + 9 h u 3 , v 2 i ⊗ N/ 4 + 4 h u 3 , v 3 i ⊗ N/ 4 + h u 3 , v 4 i ⊗ N/ 4 + h u 4 , v 1 i ⊗ N/ 4 + 3 h u 4 , v 2 i ⊗ N/ 4 + h u 4 , v 3 i ⊗ N/ 4 + h u 4 , v 4 i ⊗ N/ 4 = h u 1 , v 4 i ⊗ N/ 4 + h u 2 , v 3 i ⊗ N/ 4 + h u 2 , v 4 i ⊗ N/ 4 + h u 3 , v 2 i ⊗ N/ 4 + h u 3 , v 4 i ⊗ N/ 4 + h u 4 , v 1 i ⊗ N/ 4 + 3 h u 4 , v 2 i ⊗ N/ 4 + h u 4 , v 3 i ⊗ N/ 4 + h u 4 , v 4 i ⊗ N/ 4 . (11) All the terms in the ab ov e equatio n are zero as can b e seen by induction hypothesis either directly for h u 1 , v 4 i ⊗ N/ 4 and h u 4 , v 1 i ⊗ N/ 4 ; or using Lemma 11 for h u 2 , v 4 i ⊗ N/ 4 and h u 4 , v 2 i ⊗ N/ 4 ; or by Lemma 18 for h u 2 , v 3 i ⊗ N/ 4 and h u 3 , v 2 i ⊗ N/ 4 ; or a pplying the inclusions RM s − 1 ( m − r − 3 , m − 2 ) ⊂ RM s − 1 ( m − r − 2 , m − 2 ) ⊂ RM s − 1 ( m − r − 1 , m − 2) (see Theorem 12) as in h u 3 , v 4 i ⊗ N/ 4 , h u 4 , v 3 i ⊗ N/ 4 and h u 4 , v 4 i ⊗ N/ 4 . It remains to prove that the statement is true for t wo cases: u ∈ G s ( m − 1 , m ), v ∈ G s (0 , m ) and u ∈ G s ( m, m ), v ∈ G s ( − 1 , m ). In the first cas e v = (2 , 2 , . . . , 2) and G s ( m − 1 , m ) is an even code. In the second case v = (0 , 0 , . . . , 0). Therefore, in b oth thes e cases the statement is also true.  Now we are going to pr o ve the dua lit y relationship for the families of co des obtained by using the P lotkin constructio n. Theorem 20 F or any i nt e ger m ≥ 2 , let {RM s ( r , m ) } b e a ny fa milies of RM c o des obtaine d in The or em 10 by using the Plotkin c onstruction. Then, for e ach 0 ≤ r < m , the c o de RM s ( r , m ) is the quatern ary dual of the c o de RM s ( m − r − 1 , m ) . Pr o of: Since |RM s ( r , m ) | · |RM s ( m − r − 1 , m ) | = 2 m , it suffices to show that for any u ∈ G s ( r , m ) and v ∈ G s ( m − r − 1 , m ) we have h u , v i ⊗ N = 0. W e pro ceed by induction on m . The claim is trivia lly true for m = 2, see (4). F or even m and a n y s ≤ ⌊ m − 1 2 ⌋ , all the co des of the family RM s ( r , m ) ar e constructed b y using the Plotkin c o nstruction from the family R M s ( r , m − 1). The same ha ppens when m is o dd and s < m − 1 2 . But for m o dd and s = m − 1 2 the c o des of the family RM s ( r , m ) a re co nstructed by using the BQ- Plotkin construction fro m RM s − 1 ( r , m − 2). Hence , the initial case for the induction pro of is not only m = 1, but any m o dd and s = m − 1 2 . This sp ecific ca se w as prov ed in Theorem 19. Now, supp ose t he claim is true for the family of codes RM s ( r , m − 1), 0 ≤ r < m − 1 and 0 ≤ s ≤ ⌊ m − 2 2 ⌋ . Let u = ( u 1 | u 1 + u 2 ), where u 1 ∈ G s ( r , m − 1), u 2 ∈ G s ( r − 1 , m − 1) and v = ( v 1 | v 1 + v 2 ), where v 1 ∈ G s ( m − r − 1 , m − 1), v 2 ∈ G s ( m − r − 2 , m − 1). 20 F ro m Lemma 1 7 we have: h u , v i ⊗ N = h u 1 , v 1 i ⊗ N/ 2 + 3 h ( u 1 + u 2 ) , ( v 1 + v 2 ) i ⊗ N/ 2 = h u 1 , v 1 i ⊗ N/ 2 + 3 h u 1 , v 1 i ⊗ N/ 2 + 3 h u 1 , v 2 i ⊗ N/ 2 + 3 h u 2 , v 1 i ⊗ N/ 2 + 3 h u 2 , v 2 i ⊗ N/ 2 = 3 h u 1 , v 2 i ⊗ N/ 2 + 3 h u 2 , v 1 i ⊗ N/ 2 + 3 h u 2 , v 2 i ⊗ N/ 2 . By inductio n hypothesis , h u 1 , v 2 i ⊗ N/ 2 = 0 and h u 2 , v 1 i ⊗ N/ 2 = 0. Moreover, h u 2 , v 2 i ⊗ N/ 2 = 0, since RM s ( r − 1 , m − 1) ⊂ RM s ( r , m − 1).  W e summarize the prop erties of the RM codes in the follo wing theorem: Theorem 21 F or m ≥ 1 , t he quaternary line ar R e e d-Mul ler family of c o des {RM s ( r , m ) } , 0 ≤ s ≤ ⌊ m − 1 2 ⌋ , 0 ≤ r ≤ m , has the fol lowing pr op erties: 1. the binary length e quals n = 2 m , m ≥ 1 ; 2. the minimum distanc e is d = 2 m − r ; 3. the num b er of c o dewor ds is 2 k , wher e k = r X i =0  m i  ; 4. e ach c o de RM s ( r − 1 , m ) is a sub c o de of t he c o de RM s ( r , m ) , r > 0 . The c o de RM s (0 , m ) is t he r ep etition c o de with only one nonzer o c o dewor d (the a l l twos ve ctor). The c o de RM s ( m, m ) is the wh ole sp ac e Z 2 m − 1 4 and RM s ( m − 1 , m ) is the even c o de (i.e. the c o de with all the ve ctors of even weight); 5. the c o des RM s (1 , m ) and RM s ( m − 2 , m ) , u nder the Gr ay map, ar e a Z 4 -line ar Hadamar d and a Z 4 -line ar extende d p erfe ct c o des r esp e ctively; 6. the c o de RM s ( r , m ) is the dual c o de of the c o de RM s ( m − 1 − r, m ) for − 1 ≤ r ≤ m . In this sectio n we used everywhere the Kr onec ker inner pr oduct to de fine the duality relatio nship. But it is also p ossible to use the standard definition of inner pr o duct giv en in (1) and, in this cas e, instead o f the proper t y 6 ) in to the ab o ve Theorem, we obtain an alternative prop ert y 6’) that w e state as a new result: Theorem 22 F or m ≥ 1 and 0 ≤ s ≤ ⌊ m − 1 2 ⌋ , given a quatern ary line ar R e e d- Mul ler family {RM s ( r , m ) } of c o des, 0 ≤ r ≤ m , ther e exists a family of quaternary line ar R e e d-Mul ler c o des { RM s ( r , m ) } , monomial ly e quivalent to {RM s ( r , m ) } , such that t he c o de RM s ( r , m ) is t he dual c o de (by the standar d inner pr o duct) of RM s ( m − 1 − r , m ) for − 1 ≤ r ≤ m . Pr o of: W e hav e h u , v i ⊗ N = u · K N · v t = h u , v · K N i . Hence, we define the co de RM s ( r , m ) as the c ode ge ne r ated by a matrix G s ( r , m ), where G s ( r , m ) = G s ( r , m ) · K N . Note that the co de gener ated by the matrix G s ( r , m ) is monomi- ally eq uiv alent to t he co de generated b y G s ( r , m ).  21 5 Conclusion New constructions bas ed o n quaternary linear co des has b een prop osed such that, after doing a Gr a y map, the obtained Z 4 -linear co des fulfill the s ame pro p- erties and fundamental c har acteristics a s the binary linear R M co des. Apart from the parameters character iz ing each code an imp ortant prop erty whic h re- mains in these new presented families is that the first order RM co de is, un der the Gray map, a Z 4 -linear Hadamard code and the ( m − 2)-th or der RM co de, after the Gray map, is a Z 4 -linear extended per fect co de, lik e in the usual bina r y case. So the families of co des obtaine d in the pap er, after the Gray map, co n- tain the families of Z 4 -linear extended p erfect and Z 4 -linear Hadamard co des int ro duced in [9, 10]. 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