Z2Z4-linear codes: rank and kernel
A code C is Z2Z4-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code)…
Authors: Cristina Fern, ez-Cordoba, Jaume Pujol
1 Z 2 Z 4 -linear codes: rank and k ernel Cristina Fern ´ andez-C ´ ordoba, Jaum e P ujol and Merc ` e V illanue v a Abstract A code C is Z 2 Z 4 -additive if the set of coordin ates can b e partitione d in to two subsets X an d Y such that the pu nctured code of C by deleting the co ordinate s o utside X (r espectively , Y ) is a binar y linear cod e (respectively , a quater nary linear code) . In this p aper, the rank and dimension of the kernel for Z 2 Z 4 -linear codes, which are the co rrespond ing binary cod es of Z 2 Z 4 -additive codes, are studied. The possible values o f th ese two parameters for Z 2 Z 4 -linear c odes, gi ving lower and u pper bounds, are established. For each possible rank r between these bou nds, th e co nstruction of a Z 2 Z 4 -linear co de with rank r is g iv en. Equ iv alently , fo r each possible d imension of the kernel k , the co nstruction o f a Z 2 Z 4 -linear cod e with dimen sion of the k ernel k is g i ven. Finally , th e bound s o n the r ank, once the kernel d imension is fixed, are established an d the constructio n of a Z 2 Z 4 -additive code fo r ea ch p ossible pair ( r, k ) is gi ven. Index T erms uaternary linear cod es Z 4 -linear codes Z 2 Z 4 -additive cod es Z 2 Z 4 -linear codes kernel rank u- aternary linear cod es Z 4 -linear co des Z 2 Z 4 -additive cod es Z 2 Z 4 -linear codes kernel r ankQ I . I N T RO D U C T I O N Let Z 2 and Z 4 be the rin g of in tegers modulo 2 and modu lo 4 , respecti vely . Let Z n 2 be the set of all binary vectors o f length n and let Z n 4 be the set of all n -tuples ov er the ring Z 4 . In this paper , the e lements of Z n 4 will also be called q uaternary vectors of length n . This work was supported in part by the Spanish MEC and the European FEDER under Grants MTM2006-03 250 and TSI 2006- 14005-C02 -01. First author wishes to ackno wledge the joint sponsorship of the Fulbright Program in Spain and the Ministry of Science and Innov ation during a research stay at Auburn Univ ersit y . The material i n this paper was presented in part at t he XI International Sympos ium on P roblems of Redundanc y in Information and Control Systems, Saint Petersburg , Russia, July 2007; and at the 2nd I nternational Castle Meeting on Coding Theory and Applications, Medina del Campo, Spain, September 2008. The authors are members o f the Department of Information and Communications Engineering, Univ ersitat Aut ` onoma de Barcelona, 08193-Bellaterra, Spain. (email: { cristina.fernandez, jaume.pujol, merce.villanue v a } @autonoma.ed u) August 23, 2018 DRAFT 2 Any nonempty subset C of Z n 2 is a binary code and a subgroup of Z n 2 is called a binary linear code or a Z 2 -linear code . Equiva lently , any nonempt y subset C of Z n 4 is a quaternary code and a s ubgroup o f Z n 4 is called a quaterna ry linear code . Qu aternary li near codes can be viewe d as binary codes under the us ual Gray map defined as φ (0) = (0 , 0) , φ (1) = (0 , 1) , φ ( 2 ) = (1 , 1) , φ (3) = (1 , 0) in each coordinate. If C is a quaternary li near code, then the binary code C = φ ( C ) is called a Z 4 -linear code. The dual of a quaternary linear code C , denot ed b y C ⊥ , is called the quaternary dua l code and is defined in the standard way [19 ] in terms of the usual inner product for quaternary vectors [15]. The b inary code C ⊥ = φ ( C ⊥ ) is called the Z 4 -dual code of C = φ ( C ) . Since 1994, quaternary li near codes ha ve became significant due to its relatio nship to some classical well-known binary codes as the Nordstrom -Robinson, Kerdock, Preparata, Goeth als or Reed-Muller codes [15]. It was proved that the K erdock code and the Preparata-like code are Z 4 -linear codes and, moreover , the Z 4 -dual code of the Kerdock code is the Preparata-like code. Lately , more families of quaternary linear codes, called QRM , Z R M and RM , related to the Reed-Muller codes have been studied in [3], [4] and [25], respectively . Additive codes were first defined by Delsarte i n 1973 i n terms o f associati on s chemes [11], [12]. In general, an additive code, in a translat ion associati on scheme, is defined as a su bgroup of the underlying Abelian group. In th e special case of a binary Ham ming scheme, that i s, when the underlying Abelian group i s of o rder 2 n , t he only st ructures for th e Abel ian group are th ose of the form Z α 2 × Z β 4 , with α + 2 β = n . Therefore, the subgro ups C of Z α 2 × Z β 4 are the only additive codes in a binary Hamming scheme. In order to disti nguish them from add itive codes over finit e fields [2], we will hereafter call them Z 2 Z 4 -additive codes [5], [9], [22]. The Z 2 Z 4 - additive cod es are also inclu ded in other families of codes with an al gebraic structu re, such as mixed group codes [18] and translation i n variant propelinear codes [24]. Let C b e a Z 2 Z 4 -additive code, w hich i s a s ubgroup of Z α 2 × Z β 4 . Let Φ : Z α 2 × Z β 4 − → Z n 2 , where n = α + 2 β , be an extension of the usual Gray map give n by Φ( x, y ) = ( x, φ ( y 1 ) , . . . , φ ( y β )) for any x ∈ Z α 2 , and any y = ( y 1 , . . . , y β ) ∈ Z β 4 . This Gray m ap is an isometry which transform s Lee dist ances defined in a Z 2 Z 4 -additive code C over Z α 2 × Z β 4 to Hamm ing dist ances defined in the corresponding bi nary code C = Φ( C ) . Note that the length of C is n = α + 2 β . August 23, 2018 DRAFT 3 Giv en a Z 2 Z 4 -additive code C , the b inary code C = Φ( C ) is called a Z 2 Z 4 -linear code . Note that Z 2 Z 4 -linear codes are a generalization o f binary linear codes and Z 4 -linear codes. When β = 0 , the binary code C = C corresponds to a binary linear code. On the other hand, when α = 0 , the Z 2 Z 4 -additive code C i s a quaternary linear code and its corresponding binary code C = Φ( C ) is a Z 4 -linear code. T wo binary codes C 1 and C 2 of length n are said to be isomorphic if there exists a coordinate permutation π such that C 2 = { π ( c ) | c ∈ C 1 } . They are said to be equivalent if there e xists a vector a ∈ Z n 2 and a coordinate permut ation π such that C 2 = { a + π ( c ) | c ∈ C 1 } . T wo Z 2 Z 4 -additive codes C 1 and C 2 are said to be monomia lly equival ent , if one can be obtained from the other by permuting t he coordinates and (if necessary) changing the signs of certain Z 4 coordinates. They are said to be permutati on equivalent i f they differ onl y by a permutation of coordinates [16]. Note that if two Z 2 Z 4 -additive cod es C 1 and C 2 are mono mially equiva lent, then, after the Gray map, the correspondin g Z 2 Z 4 -linear codes C 1 = Φ( C 1 ) and C 2 = Φ( C 2 ) are isomorphic as binary codes. T wo structural properties of nonl inear bi nary codes are the rank and dimensio n of the kernel. The rank of a bin ary code C , r ank ( C ) , is simply the di mension of h C i , which is the linear span of the code words of C . The kernel of a binary code C , K ( C ) , is the set of vectors t hat leav e C in var iant under translation, i.e. K ( C ) = { x ∈ Z n 2 | C + x = C } . If C contains the all-zero vector , th en K ( C ) is a binary linear subcode of C . In general, C can be written as the union of cosets o f K ( C ) , and K ( C ) i s the lar gest such linear code for which this is true [1]. W e wi ll denote the dimensi on of t he kernel of C b y k er ( C ) . The rank and dimension of the kernel hav e been studied for som e fa milies of Z 2 Z 4 -linear codes [3], [8], [7], [17], [20], [21], [23]. These two parameters do not alwa ys give a full classification of Z 2 Z 4 -linear codes, since two n onisomorphi c Z 2 Z 4 -linear codes could hav e t he same rank and dimension of the kernel. In s pite of that, they can help in class ification, since i f two Z 2 Z 4 -linear codes ha ve d iffe rent ranks o r dimensions of the kernel, they are nonisomorphic. M oreover , in this case the correspond ing Z 2 Z 4 -additive codes are not monom ially equiv alent, so these two parameters can also help to distinguish between Z 2 Z 4 -additive codes th at are n ot monomially equiv alent . Currently , M AG M A s upports the basic fac ilities for linear codes over integer residue rings and Galois rings , and for addi tiv e codes over a finite field, which are a generalization of the linear August 23, 2018 DRAFT 4 codes over a finite field [10]. Howe ver , it does not include functions to work with Z 2 Z 4 -additive codes. For th is reason, most o f the concepts on Z 2 Z 4 -additive codes ha ve been im plemented recently as a new package in M A G M A , includ ing the com putation of the rank and kernel, and the constructio n of som e families of Z 2 Z 4 -additive codes [6]. The aim of this paper is the study of the rank and dimension of t he kernel of Z 2 Z 4 -linear codes. The paper is organized as foll ows. In Section II, we give some properties related to both Z 2 Z 4 -additive a nd Z 2 Z 4 -linear cod es, includ ing the linearity of Z 2 Z 4 -linear cod es. In Section III, we determin e all pos sible values of the rank for Z 2 Z 4 -linear codes and we prove the existence of a Z 2 Z 4 -linear code with rank r for all possible values of r . Equiva lently , in Section IV, we establish all possible values of the dimension of the kernel for Z 2 Z 4 -linear codes and we prove the existence of a Z 2 Z 4 -linear code with dimensi on of the kernel k for all possible values of k . In Section V, we determine all possi ble pairs of values ( r, k ) for which there exist a Z 2 Z 4 -linear code wi th rank r and dimensi on of the kernel k and w e const ruct a Z 2 Z 4 -linear code for any of these possib le pairs. Finally , the conclusi ons are give n in Section VI. I I . P R E L I M I N A R I E S Let C be a Z 2 Z 4 -additive cod e. Since C i s a s ubgroup o f Z α 2 × Z β 4 , i t is also i somorphic to an Abelian structure Z γ 2 × Z δ 4 . Therefore, C is of type 2 γ 4 δ as a group, it has |C | = 2 γ +2 δ code words and th e num ber of o rder two codew ords in C is 2 γ + δ . Let X (respectively Y ) be the set of Z 2 (respectiv ely Z 4 ) coordinate positions , so | X | = α and | Y | = β . Unl ess otherwis e stated, the set X correspond s t o t he first α coordin ates and Y correspon ds t o the last β coordinates. Call C X (respectiv ely C Y ) t he punctured code of C by deleting t he coordinates outsi de X (respectively Y ). Let C b be the subcode of C which contain s all order two codewords and let κ be the d imension of ( C b ) X , which is a b inary l inear code. For the case α = 0 , we will write κ = 0 . Considering all these parameters, we wi ll say that C (or equiv alently C = Φ( C ) ) is of type ( α, β ; γ , δ ; κ ) . Although a Z 2 Z 4 -additive code C is no t a free module, ev ery cod e word is uniquely expressible in the form γ X i =1 λ i u i + δ X j =1 µ j v j , where λ i ∈ Z 2 for 1 ≤ i ≤ γ , µ j ∈ Z 4 for 1 ≤ j ≤ δ and u i , v j are vectors in Z α 2 × Z β 4 of order two and four , respectively . The vectors u i , v j giv e us a generator matrix G o f size ( γ + δ ) × ( α + β ) August 23, 2018 DRAFT 5 for the code C . Moreover , we can write G as G = B 1 2 B 3 B 2 Q , (1) where B 1 , B 2 are matrices over Z 2 of size γ × α and δ × α , respectively; B 3 is a matrix ov er Z 4 of si ze γ × β with all entries in { 0 , 1 } ⊂ Z 4 ; and Q is a matrix over Z 4 of size δ × β with quaternary row vectors of o rder four . Let I n be the identi ty matrix of size n × n . In [15], it was shown that any quaternary linear code of type 2 γ 4 δ is permutation equivalent to a quaternary linear code with a generator m atrix of the form G S = 2 T 2 I γ 0 S R I δ , (2) where R , T are matrices over Z 4 with all entries in { 0 , 1 } ⊂ Z 4 , and of size δ × γ and γ × ( β − γ − δ ) , respectively; and S is a matrix over Z 4 of size δ × ( β − γ − δ ) . The fol lowing theorem i s a generalization of t his result for Z 2 Z 4 -additive codes, so it gives a canonical generator matri x for these codes. Theor em 1: [5] Let C be a Z 2 Z 4 -additive code of type ( α , β ; γ , δ ; κ ) . Then, C is permut ation equiv alent to a Z 2 Z 4 -additive code with canoni cal generator matrix of t he form G S = I κ T ′ 2 T 2 0 0 0 0 2 T 1 2 I γ − κ 0 0 S ′ S R I δ , (3) where T ′ , S ′ are matrices over Z 2 ; T 1 , T 2 , R are matrices over Z 4 with all entries in { 0 , 1 } ⊂ Z 4 ; and S is a matrix over Z 4 . The concept of d uality for Z 2 Z 4 -additive codes was also stud ied in [5], where the appropriate inner product for any two vectors u, v ∈ Z α 2 × Z β 4 was defined. Actually , in [5] it was shown that, giv en a finite Abelian group, the i nner p roduct is uni quely defined after fixing the generators in each one o f th e Abeli an element ary group s in its decomposit ion. In our case, the inner product in Z α 2 × Z β 4 is defined over Z 4 as u · v = 2( α X i =1 u i v i ) + α + β X j = α +1 u j v j ∈ Z 4 , August 23, 2018 DRAFT 6 where u, v ∈ Z α 2 × Z β 4 and the computation s are m ade taking the zeros and o nes in the first α coordinates as qu aternary zeros and ones, respectiv ely . If α = 0 , the inner product i s t he us ual one for quaternary vectors, and if β = 0 , it is twi ce the usual one for binary vectors. Then, the additive dual code of C , denot ed by C ⊥ , is defined in the standard way C ⊥ = { v ∈ Z α 2 × Z β 4 | u · v = 0 for all u ∈ C } . The corresponding b inary cod e Φ( C ⊥ ) is denot ed by C ⊥ and called the Z 2 Z 4 -dual code of C . Moreover , in [5] it was proved that the addi tiv e dual code C ⊥ , which is also a Z 2 Z 4 -additive code, is of type ( α , β ; ¯ γ , ¯ δ ; ¯ κ ) , where ¯ γ = α + γ − 2 κ, ¯ δ = β − γ − δ + κ, ¯ κ = α − κ. (4) The following two lem mas are a generalization of the same results proved for quaternary vectors and qu aternary linear codes, respectively , in [15]. Let u ∗ v denot e the component-wis e product for any u , v ∈ Z α 2 × Z β 4 . Lemma 1: For all u, v ∈ Z α 2 × Z β 4 , we have Φ( u + v ) = Φ( u ) + Φ( v ) + Φ(2 u ∗ v ) . Pr oof : Straightforward using the same ar g uments as for quaternary vectors to prove that for all u, v ∈ Z β 4 , Φ( u + v ) = Φ( u ) + Φ( v ) + Φ(2 u ∗ v ) , [15], [26]. △ Note t hat if u o r v are vectors in Z α 2 × Z β 4 of order two, then Φ( u + v ) = Φ( u ) + Φ( v ) . Lemma 2: Let C be a Z 2 Z 4 -additive code. The Z 2 Z 4 -linear code C = Φ( C ) is a b inary linear code if and only if 2 u ∗ v ∈ C for all u, v ∈ C . Pr oof : Straightforward by Lemma 1 and using the same arguments as for quaternary linear codes [15], [26]. △ Note that if G is a generator matrix o f a Z 2 Z 4 -additive code C as in (1) and { u i } γ i =1 and { v j } δ j =0 are the row vectors o f order two and four i n G , respectively , then the Z 2 Z 4 -linear code C = Φ( C ) is a bin ary l inear cod e if and onl y if 2 v j ∗ v k ∈ C , for all j, k satisfying 1 ≤ j < k ≤ δ , since the component-wi se prod uct i s bilinear . August 23, 2018 DRAFT 7 I I I . R A N K O F Z 2 Z 4 - A D D I T I V E C O D E S Let C be a Z 2 Z 4 -additive code of t ype ( α, β ; γ , δ ; κ ) and let C = Φ( C ) be the corresponding Z 2 Z 4 -linear code of bin ary length n = α + 2 β . In th is section, we will study the rank of these Z 2 Z 4 -linear codes C . W e wi ll show that there exists a Z 2 Z 4 -linear code C of type ( α , β ; γ , δ ; κ ) with r = r ank ( C ) for any possib le value of r . Lemma 3: Let C be a Z 2 Z 4 -additive code of type ( α , β ; γ , δ ; κ ) and let C = Φ( C ) be th e corresponding Z 2 Z 4 -linear code. Let G be a generator mat rix of C as in (1) and let { u i } γ i =1 be the ro ws of order two and { v j } δ j =0 the rows of order four in G . Then, h C i is generated by { Φ( u i ) } γ i =1 , { Φ( v j ) , Φ(2 v j ) } δ j =1 and { Φ(2 v j ∗ v k ) } 1 ≤ j 0 , 0 < δ + γ ≤ β + κ and κ ≤ min( α, γ ) . (5) Pr oof : Straightforward from Th eorem 1. △ Theor em 2: Let α, β , γ , δ, κ be int eger numbers satisfying (5). Then, there exists a Z 2 Z 4 -linear code C of type ( α , β ; γ , δ ; κ ) wit h r ank ( C ) = r for any r ∈ { γ + 2 δ, . . . , min( β + δ + κ, γ + 2 δ + δ 2 ) } . Pr oof : Let C be a Z 2 Z 4 -additive code of t ype ( α, β ; γ , δ ; κ ) wit h generator matrix G = I κ T ′ 0 0 0 0 0 2 T 1 2 I γ − κ 0 0 S ′ S r 0 I δ , where S r is a matrix over Z 4 of size δ × ( β − ( γ − κ ) − δ ) , and let C = Φ( C ) be its correspon ding Z 2 Z 4 -linear cod e. Let { u i } γ i =1 and { v j } δ j =0 be t he row vectors of order two and four in G , respectiv ely . August 23, 2018 DRAFT 10 By Proposit ion 1, r ank ( C ) = r = γ + 2 δ + ¯ r , where ¯ r ∈ { 0 , . . . , min( β − ( γ − κ ) − δ, δ 2 ) } . In the generator matrix G , th e Gray map image of the γ row vectors { u i } γ i =1 and the 2 δ row vectors { v j } δ j =1 , { 2 v j } δ j =1 are ind ependent binary ve ctors over Z 2 . F or each ¯ r ∈ { 0 , . . . , min( β − ( γ − κ ) − δ, δ 2 ) } , we will define S r in an appropriate way such t hat r ank ( C ) = r = γ + 2 δ + ¯ r . Let e k , 1 ≤ k ≤ δ , denote t he column vector of length δ , with a o ne in the k th coo rdinate and zeroes el sewher e. For each ¯ r ∈ { 0 , . . . , min( β − ( γ − κ ) − δ, δ 2 ) } , we can cons truct S r as a quaternary matrix w here in ¯ r column s there are ¯ r different colu mn vectors e k + e l of length δ , 1 ≤ k < l ≤ δ , and in the remaining columns there is the all -zero column vector . For each one of the ¯ r column vectors the rank increases by 1. In fact, if the column vector e k + e l is included in S r , t hen the quaternary vector 2 v k ∗ v l has only a two in the same coordinate wh ere the colum n vector e k + e l is and Φ(2 v k ∗ v l ) is independent to the vectors { Φ( u i ) } γ i =1 , { Φ( v j ) } δ j =1 , { Φ(2 v j ) } δ j =1 and { Φ(2 v s ∗ v t ) } , { s, t } 6 = { k , l } . Since the maximum number of colum ns of S r is β − ( γ − κ ) − δ and the maximum nu mber of di f ferent su ch colu mns is δ 2 , t he result fol lows. △ Let S r be a m atrix over Z 4 of size δ × ( β − ( γ − κ ) − δ ) where in ¯ r = r − ( γ + 2 δ ) columns there are ¯ r different column vectors e k + e l of l ength δ , 1 ≤ k < l ≤ δ , and i n the remaining column s there are th e all-zero colum n vector . Note that by the proo f of Theorem 2, any Z 2 Z 4 -additive code C of t ype ( α , β ; γ , δ ; κ ) wit h g enerator matrix G = I κ T ′ 0 0 0 0 0 2 T 1 2 I γ − κ 0 0 S ′ S r 0 I δ , where T ′ , T 1 and S ′ are any m atrices over Z 2 , has r ank (Φ( C )) = r = γ + 2 δ + ¯ r . Example 4: By Propositi on 1, we know that the possibl e ranks for Z 2 Z 4 -linear codes, C , of type ( α, 9; 2 , 5; 1 ) are r ank ( C ) = r ∈ { 12 , 13 , 14 , 15 } . For each p ossible r , we can construct a Z 2 Z 4 -linear code C with r ank ( C ) = r , taki ng the following generator matrix of C = Φ − 1 ( C ) : G S = 1 T ′ 0 0 0 0 0 2 T 1 2 0 0 S ′ S r 0 I 5 , August 23, 2018 DRAFT 11 where S 12 = ( 0 ) and S 13 , S 14 , and S 15 are constructed as fol lows: S 13 = 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 , S 14 = 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 , S 15 = 1 0 1 1 1 0 0 1 1 0 0 0 0 0 0 . I V . K E R N E L D I M E N S I O N O F Z 2 Z 4 - A D D I T I V E C O D E S In this section, we wil l st udy the dimension of th e kernel of Z 2 Z 4 -linear codes C = Φ( C ) . W e will also show that there exists a Z 2 Z 4 -linear code C of ty pe ( α, β ; γ , δ ; κ ) with k = k er ( C ) for any poss ible value of k . Lemma 5: Let C be a Z 2 Z 4 -additive code and let C = Φ( C ) be the correspondi ng Z 2 Z 4 -linear code. Then, K ( C ) = { Φ( u ) | u ∈ C and 2 u ∗ v ∈ C , ∀ v ∈ C } . Pr oof : By Lemm a 2 , Φ( u ) + Φ( v ) ∈ C if and only if 2 u ∗ v ∈ C for all u, v ∈ C . Thus, th e result follows. △ Note that if G is a generator matrix of a Z 2 Z 4 -additive code C and C = Φ( C ) , Φ( u ) ∈ K ( C ) if and only if u ∈ C and 2 u ∗ v ∈ C for all v ∈ G . Moreover , all code words of order two in C belong to K ( C ) . Lemma 6: Let C be a Z 2 Z 4 -additive code and let C = Φ( C ) be the correspondi ng Z 2 Z 4 -linear code. Given x, y ∈ C , Φ( x ) + Φ( y ) ∈ K ( C ) if and on ly if Φ( x + y ) ∈ K ( C ) . Pr oof : By Lemma 1, Φ( x + y + 2 x ∗ y ) = Φ( x ) + Φ( y ) . No w , by Lemma 5, Φ( x + y + 2 x ∗ y ) ∈ K ( C ) if and on ly if for all v ∈ C , 2( x + y + 2 x ∗ y ) ∗ v = 2( x + y ) ∗ v ∈ C ; that is, i f and onl y if Φ( x + y ) ∈ K ( C ) . △ Lemma 7: Let C be a Z 2 Z 4 -linear code of binary length n = α + 2 β and ty pe ( α, β ; γ , δ ; κ ) . Then, k er ( C ) ∈ { γ + δ, γ + δ + 1 , . . . , γ + 2 δ − 2 , γ + 2 δ } . Pr oof : The upper b ound γ + 2 δ comes from t he linear case. The lower bound γ + δ i s straightforward, since t here are 2 γ + δ code words of order two in C = Φ − 1 ( C ) and, by Lemma 5, the binary images by Φ of all these codew ords are in K ( C ) . Als o no te t hat i f the Z 2 Z 4 -linear August 23, 2018 DRAFT 12 code C is not linear , then th e dimension of the kernel i s equal t o or less than γ + 2 δ − 2 [21]. Therefore, k er ( C ) ∈ { γ + δ, . . . , γ + 2 δ − 2 , γ + 2 δ } . △ Giv en an int eger m > 0 , a set of vectors { v 1 , v 2 , . . . , v m } in Z α 2 × Z β 4 and a subset I = { i 1 , . . . , i l } ⊆ { 1 , . . . , m } , we denote by v I the vector v i 1 + · · · + v i l . If I = ∅ , t hen v I = 0 . Note that given I , J ⊆ { 1 , . . . , m } , v I + v J = v ( I ∪ J ) − ( I ∩ J ) + 2 v I ∩ J . Pr opos ition 2: Let C be a Z 2 Z 4 -additive code of typ e ( α, β ; γ , δ ; κ ) , with generator matri x G , and let C = Φ( C ) be the corresponding Z 2 Z 4 -linear code with k er ( C ) = γ + 2 δ − ¯ k , where ¯ k ∈ { 2 , . . . , δ } . Then, there exist a set { v 1 , v 2 , . . . , v ¯ k } of row vectors of order four i n G , such that C = [ I ⊆{ 1 ,..., ¯ k } ( K ( C ) + Φ( v I )) Pr oof : W e know th at C can be writt en as the union of cosets of K ( C ) [1]. Since | K ( C ) | = 2 γ +2 δ − ¯ k and | C | = 2 γ +2 δ , t here are exactly 2 ¯ k cosets. Let u 1 , . . . , u γ , v 1 , . . . , v δ be t he γ and δ row vectors in G of order two and four , respectively . By Lemma 5, the binary i mages by Φ of all codewords of order two are in K ( C ) . There are 2 γ + δ code words of order two generated by γ + δ codew ords. Moreover , there are δ − ¯ k code words w i of order four such that Φ( w i ) ∈ K ( C ) for all i ∈ { 1 , . . . , δ − ¯ k } , and Φ( u 1 ) , . . . , Φ( u γ ) , Φ(2 v 1 ) , . . . , Φ(2 v δ ) , Φ( w 1 ) , . . . , Φ( w δ − ¯ k ) are l inear i ndependent vectors over Z 2 . The code C can also be generated by u 1 , . . . , u γ , w 1 , . . . , w δ − ¯ k , v i 1 , . . . , v i ¯ k , where { i 1 , i 2 , . . . , i ¯ k } ⊆ { 1 , . . . , δ } . W e can assum e that v i 1 , . . . , v i ¯ k are the ¯ k row vectors v 1 , . . . , v ¯ k in G . Note t hat Φ( v I ) 6∈ K ( C ) , for any I ⊆ { 1 , . . . , ¯ k } such that I 6 = ∅ . In fact, if Φ( v I ) ∈ K ( C ) , then the set of vectors Φ( u 1 ) , . . . , Φ( u γ ) , Φ(2 v 1 ) , . . . , Φ(2 v δ ) , Φ( w 1 ) , . . . , Φ( w δ − ¯ k ) , Φ( v I ) would be linear ind ependent. Finally , we show that th e 2 ¯ k − 1 binary vectors Φ( v I ) , I ⊆ { 1 , . . . , ¯ k } and I 6 = ∅ , are in diffe rent cosets. Let Φ( v I ) and Φ( v J ) be any two of these binary vectors such that I 6 = J . If Φ( v I ) ∈ K ( C ) + Φ( v J ) , then Φ( v I ) + Φ( v J ) ∈ K ( C ) and, by Lemma 6, Φ( v I + v J ) ∈ K ( C ) . W e also hav e that v I + v J = v ( I ∪ J ) − ( I ∩ J ) + 2 v I ∩ J . Hence, Φ( v ( I ∪ J ) − ( I ∩ J ) + 2 v I ∩ J ) = Φ( v ( I ∪ J ) − ( I ∩ J ) ) + Φ(2 v I ∩ J ) ∈ K ( C ) and Φ( v ( I ∪ J ) − ( I ∩ J ) ) ∈ K ( C ) , which is a contradicti on, since ( I ∪ J ) − ( I ∩ J ) ⊆ { 1 , . . . , ¯ k } and ( I ∪ J ) − ( I ∩ J ) 6 = ∅ . △ It is important to not e that if C is a Z 2 Z 4 -linear code, then K ( C ) i s a Z 2 Z 4 -linear subcod e of C , by Lemma 6. The kernel of a Z 2 Z 4 -additive code C of type ( α, β ; γ , δ ; κ ) , denoted by August 23, 2018 DRAFT 13 K ( C ) , can be defined as K ( C ) = Φ − 1 ( K ( C )) , w here C = Φ( C ) is t he correspondin g Z 2 Z 4 -linear code. By Lemma 5, K ( C ) = { u ∈ C | 2 u ∗ v ∈ C , ∀ v ∈ C } and it is easy to see t hat K ( C ) is a Z 2 Z 4 -additive subcode of C of type ( α, β ; γ + ¯ k , δ − ¯ k ; κ ) . Note that replacing ones with twos in t he first α coordi nates, we can see Z 2 Z 4 -additive codes as quaternary linear codes. Let χ be th e m ap from Z 2 to Z 4 , wh ich i s the usual inclusion from the addi tiv e structure in Z 2 to Z 4 : χ (0) = 0 , χ (1) = 2 . This map can be extended to the m ap ( χ, I d ) : Z α 2 × Z β 4 → Z α + β 4 , which wi ll also be denoted by χ . If C is a Z 2 Z 4 -additive code o f type ( α, β ; γ , δ ; κ ) wi th generator mat rix G , then χ ( C ) is a quaternary linear code of length α + β and type 2 γ 4 δ with generator matrix G χ ( C ) = χ ( G ) . Note th at K ( C ) = χ − 1 K ( χ ( C )) and K ( χ ( C )) ⊥ is the quaternary linear code generated by the matrix H χ ( C ) 2 G χ ( C ) ∗ H χ ( C ) , where H χ ( C ) is the generator matrix of the quaternary dual code of χ ( C ) and 2 G χ ( C ) ∗ H χ ( C ) is the m atrix obt ained comput ing the compo nent-wise product 2 u ∗ v for all u ∈ G χ ( C ) , v ∈ H χ ( C ) . Moreover , by Proposition 2, giv en a Z 2 Z 4 -additive code C with generator matrix G , there exist a set { v 1 , v 2 , . . . , v ¯ k } o f row vectors of ord er four in G , such that C = [ I ⊆{ 1 ,..., ¯ k } ( K ( C ) + v I ) . Lemma 8: Let A be a symm etric m atrix over Z 2 of odd order and wit h zeroes in the m ain diagonal. Then, det( A ) = 0 . Pr oof : Let n be the order of the matrix A . The map f : Z n 2 × Z n 2 → Z n 2 defined by f ( u, v ) = uAv t is an alternating bili near form and A is a symplectic matrix [19, pp. 435]. It is known that the rank r of a symplectic matrix is always ev en [19, pp . 436]. Therefore, since the order n of A i s an odd num ber , r < n and det ( A ) = 0 . △ Pr opos ition 3: Let C be a Z 2 Z 4 -linear c ode of binary length n = α + 2 β and type ( α , β ; γ , δ ; κ ) and s = β − ( γ − κ ) − δ . Then, if s = 0 , k er ( C ) = γ + 2 δ , if s = 1 , k er ( C ) ∈ { γ + 2( δ − ⌈ δ − 1 2 ⌉ ) , . . . , γ + 2( δ − 1 ) , γ + 2 δ } , if s ≥ 2 , k er ( C ) ∈ { γ + δ, γ + δ + 1 , . . . , γ + 2 δ − 2 , γ + 2 δ } . August 23, 2018 DRAFT 14 Pr oof : For s = 0 , by Proposition 1 we hav e that r ank ( C ) = γ + 2 δ , so C i s a binary lin ear code and k er ( C ) = γ + 2 δ . F or s ≥ 2 , by Lemma 7 we ha ve that k er ( C ) ∈ { γ + δ, . . . , γ + 2 δ − 2 , γ + 2 δ } . Now , we wi ll prove the result for s = 1 . By Theorem 1, C is permutation equivalent to a Z 2 Z 4 -additive code generated by G S = I κ T ′ 2 T 2 0 0 0 0 2 T 1 2 I γ − κ 0 0 S ′ S R I δ , where S is a matrix over Z 4 of size δ × 1 . Let { u i } γ i =1 and { v j } δ j =1 be t he row vectors i n G S of order two and four , respectiv ely . If δ < 3 , then it is easy t o see that k er ( C ) = γ + 2 δ − 2 or k er ( C ) = γ + 2 δ , by Lemma 7. If δ ≥ 3 we wi ll show th at, given four vectors v j 1 , v j 2 , v j 3 , v j 4 such t hat 2 v j 1 ∗ v j 2 6∈ C and 2 v j 3 ∗ v j 4 6∈ C , then 2 v j 1 ∗ v j 2 + 2 v j 3 ∗ v j 4 ∈ C . L et e k , 1 ≤ k ≤ α + β , denote the ro w vector of length α + β , wit h a one i n the k th coordi nate and zeroes elsewhere. Then, we can write 2 v j 1 ∗ v j 2 = ( 0 , 0 , 2 c , 2 e I , 0 ) , where c ∈ { 0 , 1 } and I ⊆ { α + 2 , . . . , α + γ − κ + 1 } , and 2 v j 3 ∗ v j 4 = ( 0 , 0 , 2 c ′ , 2 e J , 0 ) , wh ere c ′ ∈ { 0 , 1 } and J ⊆ { α + 2 , . . . , α + γ − κ + 1 } . W e denote b y u I (resp. u J ) the row vector obt ained by adding the row vectors of o rder two in G S with 2 in the coordinate positions given by I (resp. J ). Then, u I = ( 0 , 0 , 2 d, 2 e I , 0 ) ∈ C with d ∈ { 0 , 1 } (resp. u J = ( 0 , 0 , 2 d ′ , 2 e J , 0 ) ∈ C with d ′ ∈ { 0 , 1 } ). Since 2 v j 1 ∗ v j 2 6∈ C (resp. 2 v j 3 ∗ v j 4 6∈ C ) we hav e 2 v j 1 ∗ v j 2 = u I + ( 0 , 0 , 2 , 0 , 0 ) (resp. 2 v j 3 ∗ v j 4 = u J + ( 0 , 0 , 2 , 0 , 0 ) ). Therefore, 2 v j 1 ∗ v j 2 + 2 v j 3 ∗ v j 4 = u I + u J ∈ C By Proposition 2, there exist ¯ k row vectors v 1 , v 2 , . . . , v ¯ k in G S , s uch that Φ( v I ) 6∈ K ( C ) for any nonempty sub set I ⊆ { 1 , . . . , ¯ k } and k er ( C ) = γ + 2 δ − ¯ k . Assume ¯ k is odd. W e wi ll show that there exists a subset I ⊆ { 1 , . . . , ¯ k } such that Φ( v I ) ∈ K ( C ) . Since thi s is a contradiction, ¯ k can not be an odd num ber and the assertion wil l be proved. By Lemma 5, in order to prove that there exists I ⊆ { 1 , . . . , ¯ k } such that Φ( v I ) ∈ K ( C ) , it is enough to prove that 2 v I ∗ v j ∈ C for all j ∈ { 1 , . . . , ¯ k } . Th at is , 2 v i ∗ v j ∈ C for all i ∈ I and j ∈ { 1 , . . . , ¯ k } or , following the above remark, for each j ∈ { 1 , . . . , ¯ k } the num ber of i ∈ I such that 2 v i ∗ v j 6∈ C i s even. W e define a sym metric m atrix A = ( a ij ) , 1 ≤ i, j ≤ ¯ k , in the following way: a ij = 1 i f 2 v i ∗ v j 6∈ C and 0 otherwise. Therefore, A is a sym metric matrix of August 23, 2018 DRAFT 15 odd order and with zeroes in the main diagonal. Lemma 8 s hows that det( A ) = 0 and hence there exists a linear combination of s ome ro ws, i 1 , . . . , i l , of A equal to 0 . The vector Φ( v I ) , where I = { i 1 , . . . , i l } , b elongs to K ( C ) . This compl etes t he proof. △ Example 5: Continuing with Example 1, the dim ension of the kernel for a Hadam ard Z 4 - linear code H was computed in [23] and [17] and the dimens ion of t he kernel for an extended 1-perfect Z 4 -linear code C in [7]. Specifically , k er ( H ) = γ + δ + 1 if δ ≥ 3 γ + 2 δ if δ = 1 , 2 and k er ( C ) = ¯ γ + ¯ δ + 1 if δ ≥ 3 ¯ γ + ¯ δ + 2 if δ = 2 ¯ γ + ¯ δ + t if δ = 1 . Example 6: Continuing w ith Example 2, th e dimensio n of the kernel for a Hadamard Z 2 Z 4 - linear code H was computed i n [23] and the dimension of the kernel for an extended 1-perfect Z 2 Z 4 -linear code C in [7]. Specifically , k er ( H ) = γ + δ if δ ≥ 2 γ + 2 δ if δ = 0 , 1 and k er ( C ) = ¯ γ + ¯ δ + 1 if δ ≥ 1 ¯ γ + 2 ¯ δ if δ = 0 . Note that the kernel dimensi on of th e Hadamard Z 2 Z 4 -linear codes satis fies the lower bound. Example 7: Let QRM ( r , m ) be the class of Z 4 -linear Reed-Muller codes defined in [3], as in Example 3. The dimension of the kernel of any cod e C ∈ QRM ( r , m ) is k er ( C ) = r X i =0 m i + 1 = δ + 1 , except for r = m (in thi s case, C = Z 2 m +1 2 ), [3]. Therefore, Z 4 -linear Kerdock-like codes and e xtended Z 4 -linear Preparata-like codes of binary length 4 m hav e dimension o f the kernel k er ( K ) = 2 m + 1 and k er ( P ) = 2 2 m − 1 − 2 m + 1 , respectiv ely [3], [8]. August 23, 2018 DRAFT 16 As in Section III for the rank, the n ext poin t to be solved here is ho w to construct Z 2 Z 4 -linear codes with any dimension of the kernel in th e range of possibil ities given by Proposition 3. Theor em 3: Let α, β , γ , δ, κ be int eger numbers satisfying (5). Then, there exists a Z 2 Z 4 -linear code C of type ( α , β ; γ , δ ; κ ) wit h k er ( C ) = k for any k ∈ { γ + δ , . . . , γ + 2 δ − 2 , γ + 2 δ } if s ≥ 2 { γ + 2( δ − ⌈ δ − 1 2 ⌉ ) , . . . , γ + 2( δ − 1) , γ + 2 δ } if s = 1 { γ + 2 δ } if s = 0 , where s = β − ( γ − κ ) − δ. Pr oof : Let C be a Z 2 Z 4 -additive code of t ype ( α, β ; γ , δ ; κ ) wit h generator matrix G = I κ T ′ 0 0 0 0 0 0 2 I γ − κ 0 0 S ′ S k 0 I δ , where S k is a matrix over Z 4 of si ze δ × s , and let C = Φ( C ) be i ts corresponding Z 2 Z 4 - linear code. T aking S k as the all-zero matrix over Z 4 , the code C is a binary li near code, so k er ( C ) = k = γ + 2 δ . When s = 1 , for each ¯ k ∈ { 2 , 4 , . . . , 2 ⌈ δ − 1 2 ⌉} and k = γ + 2 δ − ¯ k , we can construct a matrix S k over Z 4 of size δ × 1 with an e ven num ber of on es, ¯ k , and zeroes else where. In this case, k er ( C ) = k = γ + 2 δ − ¯ k , by t he proof of Propositi on 3. Finally , when s ≥ 2 , for each ¯ k ∈ { 2 , 3 , . . . , δ } and k = γ + 2 δ − ¯ k , we can construct a matrix S k over Z 4 of size δ × s , such that only in the l ast δ − ¯ k row vectors all com ponents are zero and, moreover , i n the first ¯ k coo rdinates of each column vector there are an e ven number of ones and zeros elsewhere. In t his case, b y the s ame arguments as in the proof of Propositi on 3, it is easy to p rove t hat k er ( C ) = k = γ + 2 δ − ¯ k . △ Example 8: By Proposition 3, we k now that th e possibl e di mensions of the kernel for Z 2 Z 4 - linear codes, C , of type ( α, 9; 2 , 5; 1) are k er ( C ) = k ∈ { 12 , 10 , 9 , 8 , 7 } . For each pos sible k , we can construct a Z 2 Z 4 -linear code C with k er ( C ) = k , t aking the following generator matrix of C = Φ − 1 ( C ) : G S = 1 T ′ 0 0 0 0 0 0 2 0 0 S ′ S k 0 I 5 , August 23, 2018 DRAFT 17 where S 12 = ( 0 ) and S 10 , S 9 , S 8 and S 7 are constructed as follows: S 10 = 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 , S 9 = 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 , S 8 = 1 0 0 1 1 0 1 1 0 1 0 0 0 0 0 , S 7 = 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 . V . P A I R S O F R A N K A N D K E R N E L D I M E N S I O N O F Z 2 Z 4 - A D D I T I V E C O D E S In this section, once the di mension of the kernel is fixed, lower and upper bound s on the rank are establis hed. W e will show that there exists a Z 2 Z 4 -linear code C of type ( α, β ; γ , δ ; κ ) wit h r = r ank ( C ) and k = k er ( C ) for any possibl e pair of values ( r, k ) . Lemma 9: Let C be a Z 2 Z 4 -additive code of type ( α , β ; γ , δ ; κ ) and let C = Φ( C ) be th e corresponding Z 2 Z 4 -linear code. If r ank ( C ) = γ + 2 δ + ¯ r and k er ( C ) = γ + 2 δ − ¯ k , with ¯ k ≥ 2 , then 1 ≤ ¯ r ≤ ¯ k 2 . Pr oof : There exist { u i } γ i =1 and { v j } δ j =1 vectors of order two and four respectively , such that they generate the code C and C = S I ⊆{ 1 ,..., ¯ k } ( K ( C ) + Φ( v I )) by Proposition 2. Note t hat Φ( v j ) ∈ K ( C ) if and only if j ∈ { ¯ k + 1 , . . . , δ } . By Lemma 5, for all j ∈ { ¯ k + 1 , . . . , δ } and i ∈ { 1 , . . . , δ } , as Φ( v j ) ∈ K ( C ) , 2 v j ∗ v i ∈ C and, cons equently , Φ(2 v j ∗ v i ) i s a linear combination of { Φ( u i ) } γ i =1 and { Φ(2 v j ) } δ j =1 . As a result, h C i is g enerated by { Φ( u i ) } γ i =1 , { Φ( v j ) , Φ(2 v j ) } δ j =1 and { Φ(2 v t ∗ v s ) } 1 ≤ s
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