Survey onZ2Z4-additive codes1
J. Borges⋆, C. Fern´andez-C´ordoba⋆, J. Pujol⋆, J. Rif`a⋆, M. Villanueva⋆
⋆Department of Information and Communications Engineering, Universitat Aut`onoma de Bar- celona, 08193-Bellaterra, Spain.
Email:{joaquim.borges, cristina.fernandez, jaume.pujol, josep.rifa, merce.villanueva}@uab.cat
Abstract
A codeCisZ2Z4-additive if the set of coordinates can be partitioned into two subsetsXand Y such that the punctured code ofCby deleting the coordinates outsideX(respectively,Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes ofZ2Z4-additive codes under an extended Gray map are calledZ2Z4-linear codes, which seem to be a very distinguished class of binary group codes.
As for binary and quaternary linear codes, for these codes the fundamental parameters are shown and standard forms for generator and parity-check matrices are given, defining the appropriate concept of duality. The main results onZ2Z4-additive self-dual andZ2Z4- additive formally self-dual codes are also presented, as well as, the results on the invariants rank and dimension of the kernel for these codes are given. Several families of important binary codes fall in the class of Z2Z4-linear codes. In this survey, we review character- izations, properties and constructions of perfect and extended perfectZ2Z4-linear codes, Hadamard Z2Z4-linear codes, Reed-Muller Z2Z4-linear codes, maximum distance sepa- rableZ2Z4-linear codes, and Preparata-like and Kerdock-likeZ2Z4-linear codes. Finally, applications ofZ2Z4-additive codes to steganography are also presented.
1. Introduction
LetZ2 andZ4 be the ring of integers modulo 2 and 4 respectively. LetZn2 denote the set of all binary vectors of lengthnand letZn4 be the set of alln-tuples over the ringZ4. In this paper, the elements ofZn4 will also be called quaternary vectors of lengthn. We denote by0ℓ and1ℓ the all-zero and the all-one vectors, respectively, of lengthℓ. If the length of such vectors is clear from the context we omit the parameterℓ.
1This work has been partially supported by the Spanish MICINN under Grants TIN2010-17358 and TIN2013- 40524-P, and by the Catalan AGAUR under Grant 2014SGR-691. The authors are in alphabetical order.
The Hamming distancedH(u, v) between two vectors u, v ∈ Zn2 is the number of coordi- nates in whichuandvdiffer, and theHamming weightof a vectoru∈Zn2, denoted bywH(u), is the number of nonzero coordinates of u, so dH(u, v) = wH(u −v). On the other hand, the Lee weights over the elements in Z4 are defined as: wL(0) = 0, wL(1) = wL(3) = 1, wL(2) = 2. Then, the Lee weightof a vectoru ∈ Zn4, denoted bywL(u), is the addition of the weights of its coordinates, whereas theLee distancedL(u, v)between two vectorsu, v ∈Zn4 is dL(u, v) = wL(u−v).
Any nonempty subset C of Zn2 is a binary code and a subgroup of Zn2 is called a binary linear code or a Z2-linear code. Equivalently, any nonempty subsetC of Zn4 is a quaternary code and a subgroup ofZn4 is called aquaternary linear code. The elements of a code are called codewords. If C is a binary linear code, it is isomorphic to an additive group Zk2, so C has dimensionkand it has2kcodewords. Equivalently, ifC is a quaternary linear code, since it is a subgroup ofZn4, it is isomorphic to an abelian structureZγ2 ×Zδ4. Therefore, we have thatC is of type2γ4δ as a group, and it has|C| = 2γ+2δcodewords.
Quaternary codes can be viewed as binary codes under the usual Gray map defined asϕ(0) = (0,0), ϕ(1) = (0,1), ϕ(2) = (1,1), ϕ(3) = (1,0)in each coordinate. IfCis a quaternary linear code, then the binary codeC =ϕ(C)is called aZ4-linearcode. The dual of a quaternary linear codeC, denoted byC⊥, is called thequaternary dual codeand is defined in the standard way [41]
in terms of the usual inner product for quaternary vectors [33]. The binary codeC⊥=ϕ(C⊥)is called theZ4-dual codeofC =ϕ(C).
The minimum Hamming distance dH(C) of a binary code C is the minimum value of dH(u, v) for u, v ∈ C satisfying u ̸= v. The minimum Hamming weight of a binary code C, denoted bywH(C), is the minimum value ofwH(u)foru∈C\{0}. It is well known that if Cis a binary linear code,dH(C) =wH(C)[41]. Equivalently, theminimum Lee distancedL(C) of a quaternary code C is the minimum value of dL(u, v)for u, v ∈ C satisfyingu ̸= v. The minimum Lee weightof a quaternary codeC, denoted bywL(C), is the minimum value ofwL(u) foru ∈ C\{0}. Again, if C is a quaternary linear code, dL(C) = wL(C). Note that the Gray map ϕ is an isometry which transforms Lee distances over Zn4 into Hamming distances over Z2n2 . Therefore, the minimum Lee weight of a quaternary codeC coincides with the minimum Hamming weight ofC =ϕ(C), that iswL(C) =wH(ϕ(C)).
Since 1994, quaternary linear codes have been studied and became significant since, after applying the Gray map, we obtain binary nonlinear codes better than any known binary lin- ear code with the same parameters. More specifically, Hammons et. al. [33, 64] show how to construct well known binary nonlinear codes like the Nordstrom-Robinson code, Kerdock codes and Delsarte-Goethals codes asZ4-linear codes, that is, as the Gray map image of qua- ternary linear codes. Furthermore, they solve an old open problem on coding theory about that the Hamming weight enumerators of the nonlinear Kerdock and Preparata codes satisfy the MacWilliams identities. Actually, they prove that the Kerdock codes and some Preparata-like codes areZ4-linear codes and, moreover, theZ4-dual code of the Kerdock code is a Preparata- like code. Later, several other Z4-linear codes with the same parameters as some well known families of binary linear codes (for example, extended Hamming, Hadamard, and Reed-Muller codes) have been studied and classified [8, 9, 15, 14, 38, 43, 44, 48, 50, 63].
Additive codes were first defined by Delsarte in1973in terms of association schemes [22, 23]. According to this definition, an additive code is a subgroup of the underlying abelian group in a translation association scheme. On the other hand, translation invariant propelinear
codes were first defined in1997[55, 58], where it is proved that they are group-isomorphic to subgroups ofZα2 ×Zβ4 ×Qσ8, beingQ8 the non-abelian quaternion group on eight elements. In the special case of a binary Hamming scheme, that is, when the underlying abelian group is of order 2n, the additive codes coincide with the abelian translation invariant propelinear codes.
Hence, as it is pointed out in [23, 54], the only structures for the abelian group are those of the formZα2×Zβ4, withα+2β =n. Therefore, the codes that are subgroups ofZα2×Zβ4 are the only additive codes in the binary Hamming scheme. In order to distinguish them from additive codes over finite fields [4], from now on, we will call themZ2Z4-additive codes. Note that one could think of other families of codes with an algebraic structure that also include theZ2Z4-additive codes, such as mixed group codes [13, 34, 40].
SinceZ2Z4-additive codes are subgroups ofZα2 ×Zβ4, they can be seen as a generalization of binary (whenβ = 0) and quaternary (whenα = 0) linear codes. As for quaternary linear codes, after applying the Gray map to the Z4 coordinates of a Z2Z4-additive code, we obtain binary codes calledZ2Z4-linear codes. There areZ2Z4-linear codes in several important classes of binary codes. For example, Z2Z4-linear perfect single error-correcting codes (or 1-perfect codes) are found in [55] and fully characterized in [16]. Also, in subsequent papers [14, 38, 47, 48], Z2Z4-linear extended perfect and Hadamard codes are studied and classified. Note that Z2Z4-additive codes have allowed to classify more binary nonlinear codes, giving them a structure as Z2Z4-additive codes. Although it is not easy to determine whether a code has a Z2Z4-additive structure, and whether it is unique or not, it seems that there are many more Z2Z4-linear codes than linear. In this sense, recently, a preliminary proposal about counting Z2Z4-additive codes can be found in [26]. Finally, mention that a permutation decoding method forZ2Z4-linear codes is described in [3].
Part of the research developed by the Combinatorics, Coding and Security Group (CCSG) deals with quaternary linear codes, as well asZ2Z4-additive codes. Since there is not any sym- bolic software to work withZ2Z4-additive codes, the members of CCSG have been developing a new package [12, 46, 52] in MAGMA [19] that supports the basic facilities for these codes.
Specifically, this new MAGMApackage generalizes most of the functions for quaternary linear codes to Z2Z4-additive codes, and includes new functions specific for these kind of codes. A beta version of this package and the manual with the description of all functions can be down- loaded from the web pagehttp://ccsg.uab.cat(for any comment or further information about this package, you can send an e-mail tosupport-ccsg@deic.uab.cat).
The aim of this paper is to give a complete survey on Z2Z4-additive codes and their corre- spondingZ2Z4-linear codes. It is organized as follows. In Section 2, we recall the definition of aZ2Z4-additive andZ2Z4-linear code, and we describe their fundamental parameters and a standard form for the generator matrices of these codes. Section 3 is devoted to the duality con- cept forZ2Z4-additive codes defining the appropriate inner product, showing how the generator and parity-check matrices are related, as well as how the parameters of the dual code can be computed from the parameters of the code. In Section 4, we discuss aboutZ2Z4-additive self- dual codes. Separability, antipodality and Type of these codes are studied. Moreover, we give different constructions of such codes. Briefly, we also discuss about formally self-dual additive codes. In Section 5, we present the possible values of two invariants: the rank and dimension of the kernel, forZ2Z4-additive codes. Such techniques are further applied to specific families ofZ2Z4-additive codes. In Section 6, constructions, classification and properties like rank and
dimension of the kernel are established for several families of Z2Z4-additive codes. An inter- esting application ofZ2Z4-additive codes to steganography is presented in Section 7. Finally, in Section 8, we give some conclusions and discuss about further research on these codes.
2. Z2Z4-additive codes
In this section, we recall some definitions and concepts related toZ2Z4-additive codes. We also describe their fundamental parameters and a standard form for the generator matrices of these codes. The material of this section is a summary of the results presented in [10, 11].
From now on, we focus onZ2Z4-additive codes C, which are subgroups of Zα2 ×Zβ4. The Z2Z4-additive codes can also be seen as binary codes, calledZ2Z4-linear codes, by considering the extension of the usual Gray map:Φ :Zα2 ×Zβ4 −→Zn2, wheren =α+ 2β, given by
Φ(x, y) = (x, ϕ(y1), . . . , ϕ(yβ))
∀x∈Zα2, ∀y= (y1, . . . , yβ)∈Zβ4;
where ϕ : Z4 −→ Z22 is the usual Gray map, that is, ϕ(0) = (0,0), ϕ(1) = (0,1), ϕ(2) = (1,1), ϕ(3) = (1,0).For a vectorv = (v1, v2)∈Zα2 ×Zβ4, we define the weight ofv, denoted byw(v), aswH(v1) +wL(v2). Note that sincew(v) =wH(Φ(v)), the Gray mapΦis an isom- etry which transforms distances defined in aZ2Z4-additive codeC over Zα2 ×Zβ4 to Hamming distances defined in the corresponding Z2Z4-linear code C = Φ(C). Note that the length of C = Φ(C)isn =α+ 2β.
Let C be aZ2Z4-additive code. Since C is a subgroup of Zα2 ×Zβ4, it is isomorphic to an abelian structure Zγ2 ×Zδ4. Therefore, C is of type 2γ4δ as a group, and it has |C| = 2γ+2δ codewords. Let X (respectivelyY) be the set ofZ2 (respectivelyZ4) coordinate positions, so
|X|=αand|Y|=β. Unless otherwise stated, the setX corresponds to the firstαcoordinates andY corresponds to the lastβ coordinates. CallCX (respectivelyCY) the punctured code of C by deleting the coordinates outside X (respectively Y). Let Cb be the subcode of C which contains all order two codewords and letκbe the dimension of(Cb)X, which is a binary linear code. For the caseα = 0, we writeκ = 0. Considering all these parameters, we say that the Z2Z4-additive code C (or equivalently the corresponding Z2Z4-linear code C = Φ(C)) is of type(α, β;γ, δ;κ).
Note that CY is a quaternary linear code of type (0, β;γY, δ; 0), where 0 ≤ γY ≤ γ, and CX is a binary linear code of type (α,0;γX,0;γX), where κ ≤ γX ≤ κ+δ. Note also that Z2Z4-linear codes are a generalization of binary linear codes andZ4-linear codes. Whenβ = 0, the binary code C = C corresponds to a binary linear code. On the other hand, when α = 0, theZ2Z4-additive code C is a quaternary linear code and the correspondingZ2Z4-linear code C = Φ(C)is aZ4-linear code.
Two Z2Z4-additive codes C1 and C2 both of type (α, β;γ, δ;κ) are said to be monomially equivalent, if one can be obtained from the other by permutating the coordinates and (if nec- essary) changing the signs of certainZ4 coordinates. TwoZ2Z4-additive codes are said to be permutation equivalentif they differ only by a permutation of coordinates. If twoZ2Z4-additive
codesC1andC2 are monomially equivalent, then, after the Gray map, the correspondingZ2Z4- linear codesC1 = Φ(C1)andC2 = Φ(C2)are isomorphic as binary codes. Note that the inverse statement is not always true. The monomial automorphism groupof aZ2Z4-additive codeC, denoted by MAut(C), is the group generated by all permutations and sign-changes of theZ4co- ordinates that preserves the set of codewords ofC, while thepermutation automorphism group ofC, denoted by PAut(C), is the group generated by all permutations that preserves the set of codewords ofC [36].
AlthoughCis not a free module, every codeword is uniquely expressible in the form
∑γ i=1
λiu(i)+
∑γ+δ j=γ+1
µjv(j),
whereλi ∈ Z2 for 1 ≤ i ≤ γ, µj ∈ Z4 forγ + 1 ≤ j ≤ γ +δ andu(i), v(j) are vectors in Zα2 ×Zβ4 of order two and order four, respectively. Moreover, the vectors u(i), v(j) give us a generator matrixG of size (γ +δ)×(α+β)for the code C. This generator matrixG can be written as
G=
( B1 2B3 B2 Q
)
, (1)
whereB1, B2 are matrices overZ2 of sizeγ ×αandδ×α, respectively; B3 is a matrix over Z4 of sizeγ×β with all entries in{0,1} ⊂ Z4; andQis a matrix overZ4 of sizeδ×β with quaternary row vectors of order four.
Let Ik be the identity matrix of size k×k. In [33, 64], it was shown that any quaternary linear code of type (0, β;γ, δ; 0) is permutation equivalent to a quaternary linear code with a generator matrix of the form
GS =
( 2T 2Iγ 0 S R Iδ
)
, (2)
whereR, T are matrices overZ4with all entries in{0,1} ⊂Z4, of sizeδ×γandγ×(β−γ−δ), respectively; and S is a matrix overZ4 of size δ×(β −γ −δ). In [10, 11], this result was generalized forZ2Z4-additive codes as follows:
Theorem 2.1 [10, 11] LetC be aZ2Z4-additive code of type(α, β;γ, δ;κ). Then,C is permu- tation equivalent to aZ2Z4-additive code with canonical generator matrix of the form
GS =
Iκ Tb 2T2 0 0 0 0 2T1 2Iγ−κ 0 0 Sb Sq R Iδ
, (3)
whereTb, Sb are matrices overZ2; T1, T2, R are matrices overZ4 with all entries in {0,1} ⊂ Z4; andSqis a matrix overZ4.
Example 2.2 LetC be aZ2Z4-additive code of type(3,4; 3,1; 3)with generator matrix
G=
1 0 0 2 2 0 0 1 1 1 2 2 2 2 1 1 0 2 2 0 0 1 1 1 1 1 1 1
.
By Theorem 2.1,C is permutation equivalent to aZ2Z4-additive code with canonical generator matrix
GS =
1 0 0 2 2 0 0 0 1 0 0 0 0 0 0 0 1 2 2 0 0 0 0 0 1 1 1 1
. (4)
3. Duality ofZ2Z4-additive codes
For linear codes over finite fields or finite rings, there exists the well known concept of duality.
In this section, we explain the duality forZ2Z4-additive codes, taking advantage of their abelian group structure. We also show how to compute the type of an additive dual code, and how to construct a parity-check matrix of aZ2Z4-additive code, or equivalently a generator matrix of its additive dual code, when the code is generated by a canonical generator matrix as in (3). The material of this section is a summary of the results presented in [10, 11].
The appropriate inner product of two vectorsu= (u1, u2, . . . , uα+β),v = (v1, v2, . . . , vα+β) inZα2 ×Zβ4 is given by [10, 11]
⟨u, v⟩= 2(
∑α i=1
uivi) +
α+β∑
j=α+1
ujvj ∈ Z4,
as usual, assuming that the first α coordinates are binary and the last β coordinates are qua- ternary. Note that the computations are made considering the zeros and ones in theα binary coordinates as quaternary zeros and ones, respectively. We refer to this product as thestandard inner product, that can also be written as
⟨u, v⟩=u·Jn·vt, where Jn =
( 2Iα 0 0 Iβ
)
is a diagonal matrix over Z4. Note that when α = 0 the inner product is the usual one for quaternary vectors, and when β = 0 it is twice the usual one for binary vectors.
LetC be aZ2Z4-additive code of type(α, β;γ, δ;κ) and letC = Φ(C)be the corresponding Z2Z4-linear code. Theadditive orthogonal codeofC, denoted byC⊥, is defined in the standard way as
C⊥ ={v ∈Zα2 ×Zβ4 | ⟨u, v⟩= 0for allu∈ C}.
We also callC⊥ theadditive dual codeofC. The corresponding binary codeΦ(C⊥)is denoted byC⊥ and calledZ2Z4-dual codeofC. In the case thatα = 0, that is, when C is a quaternary linear code,C⊥is also called thequaternary dual codeofC andC⊥theZ4-dual codeofC. The additive dual codeC⊥is also aZ2Z4-additive code, that is, a subgroup ofZα2 ×Zβ4.
LetC be aZ2Z4-additive code of type(α, β;γ, δ;κ). Theweight enumeratorofC is WC(x, y) = ∑
c∈C
xn−w(c)yw(c), (5)
wheren = α+ 2β andw(c) stands for the Lee weight of a codewordc ∈ C. We know from [10, 11, 23, 55] that for the weight enumerator defined in (5) we have that
WC⊥(x, y) = 1
|C|WC(x+y, x−y). (6) Therefore, taking x = y, we obtain that |C||C⊥| = 2n. Note that the weight distribution of a Z2Z4-additive code C refers to the Lee weight, which coincides with the Hamming weight of the corresponding Z2Z4-linear code C = Φ(C). The codes C and C⊥ are not necessarily linear, so they are not dual in the binary linear sense, but the weight enumerator ofC⊥ is the MacWilliams transform of the weight enumerator ofC. This remarkable relationship was first established for the specific case ofZ4-linear codes in [33, 64], where it is pointed out that the Kerdock code is the quaternary dual of some Preparata-like code.
Note that one could think onZ2Z4-additive codes (orZ2Z4-linear codes) just as quaternary linear codes (orZ4-linear codes), replacing the ones with twos in the binary coordinates. How- ever, in this case, the corresponding quaternary linear codes of anZ2Z4-additive codeC and its additive dual codeC⊥are not necessarily quaternary dual codes. Take, for example,α=β = 1 and the vectors v = (1,3) and w = (1,2). It is easy to check that ⟨v, w⟩ = 0, so v and w are orthogonal. However, if we replace the ones with twos in the binary coordinates of these vectors, we obtainv′ = (2,3)andw′ = (2,2), which are not orthogonal in the quaternary sense.
The next results were established in [10, 11], where the computation of the type of the addi- tive dual codeC⊥of a givenZ2Z4-additive codeC, as well as the construction of the generator matrix of C⊥ in terms of the generator matrix of C are given. Previously, in [33, 64], it was shown that ifC is a quaternary linear code of type(0, β;γ, δ; 0), then the quaternary dual code C⊥is of type(0, β;γ, β−γ−δ; 0). Moreover, ifC has canonical generator matrix (2), then the generator matrix ofC⊥is
HS =
( 0 2Iγ 2Rt Iβ−γ−δ Tt −(S+RT)t
)
, (7)
whereR, T are matrices overZ4with all entries in{0,1} ⊂Z4of sizeδ×γandγ×(β−γ−δ), respectively; andS is a matrix over Z4 of sizeδ×(β −γ −δ). In [10, 11], these two results were generalized forZ2Z4-additive codes as follows:
Theorem 3.1 [10, 11] LetC be aZ2Z4-additive code of type(α, β;γ, δ;κ). The additive dual codeC⊥is then of type(α, β; ¯γ,δ; ¯¯ κ), where
¯
γ =α+γ−2κ, δ¯=β−γ−δ+κ,
¯
κ=α−κ.
Theorem 3.2 [10, 11] Let C be a Z2Z4-additive code of type (α, β;γ, δ;κ) with canonical generator matrix (3). Then, the generator matrix ofC⊥is
HS =
Tbt Iα−κ 0 0 2Sbt 0 0 0 2Iγ−κ 2Rt T2t 0 Iβ+κ−γ−δ T1t −(
Sq+RT1)t
, (8)
whereTb, T2 are matrices overZ2;T1, R, Sb are matrices overZ4 with all entries in {0,1} ⊂ Z4; andSqis a matrix overZ4.
Note that by Theorems 3.1 and 3.2, if C is aZ2Z4-additive code of type(α, β;γ, δ;κ)with canonical generator matrix (3), thenC⊥ is permutation equivalent to aZ2Z4-additive code with canonical generator matrix
I¯κ Tbt 2Sbt 0 0 0 0 2Rt 2Iγ¯−¯κ 0 0 T2t −(
Sq+RT1)t
T1t Iδ¯
, (9)
whereTb, T2are matrices overZ2;T1, R, Sbare matrices overZ4with all entries in{0,1} ⊂Z4; andSqis a matrix overZ4. Moreover,γ¯=α+γ−2κ,δ¯=β−γ−δ+κandκ¯=α−κ.
Finally, it is also worth mentioning that a generator matrix of C⊥ can be seen as a parity- check matrix forC. Analogously, by linearity, we can use a generator matrix ofC as a parity- check matrix for C⊥. Therefore, Theorem 3.2 also shows how to construct the parity-check matrix of aZ2Z4-additive code generated by a canonical generator matrix as in (3).
Example 3.3 Let CS be a Z2Z4-additive code of type (3,4; 3,1; 1) with canonical generator matrix (4) given in Example 2.2. By Theorems 3.1 and 3.2, the additive dual codeCS⊥is of type (3,4; 0,3; 0)and has generator matrix
HS =
1 0 1 1 0 0 3 1 0 1 0 1 0 3 0 0 0 0 0 1 3
.
4. Z2Z4-additive self-dual codes
AZ2Z4-additive codeC isself-orthogonalifC⊥ ⊆ C, and it isself-dualifC =C⊥. Self-duality for binary and quaternary linear codes has been extensively studied. For the quaternary case, the Gray map images of quaternary self-dual codes are also very interesting since they arefor- mally self-dual, that is, their Hamming weight enumerators are invariant under the MacWilliams transform [33]. Therefore, a next logical step is to studyZ2Z4-additive self-dual codes and their Gray map images. The material in this chapter is a summary of the results presented in [7, 25].
4.1. Classification ofZ2Z4-additive self-dual codes
In this subsection, we show thatZ2Z4-additive self-dual codes can be characterized in terms of some properties such us separability, antipodality and Type. Moreover, we determine all the possible values for the parameters α and β of a Z2Z4-additive self-dual code, as well as the weight enumerator of such codes.
4.1.1. SEPARABILITY AND ANTIPODALITY
The following proposition determine the type of a Z2Z4-additive self-dual code taking into account the relation of the parameters given in Theorem 3.1.
Proposition 4.1 [7] IfC is aZ2Z4-additive self-dual code, then C is of type (2κ, β;β +κ− 2δ, δ;κ),|C| = 2κ+β,|Cb|= 2κ+β−δ, and(Cb)X is a binary self-dual code.
LetC be aZ2Z4-additive code. IfC =CX×CY, thenC is calledseparable. IfCis separable, then the generator matrix ofC in standard form is
GS =
Iκ Tb 0 0 0 0 0 2T1 2Iγ−κ 0 0 0 Sq R Iδ
.
The following theorem show some properties of separableZ2Z4-additive self-dual codes.
Theorem 4.2 [7] LetC be aZ2Z4-additive self-dual code of type(2κ, β;β+κ−2δ, δ;κ). The following statements are equivalent:
(i) CX is a binary self-orthogonal code.
(ii) CX is a binary self-dual code.
(iii) |CX|= 2κ.
(iv) CY is a quaternary self-orthogonal code.
(v) CY is a quaternary self-dual code.
(vi) |CY|= 2β. (vii) C is separable.
From the above theorem, ifC is aZ2Z4-additive self-dual code, thenCX is binary self-dual if and only ifCY is quaternary self-dual. Moreover, if C is a separableZ2Z4-additive code, CX
is binary self-dual andCY is quaternary self-dual, thenC is also self-dual, as it is stated in the following theorem.
Theorem 4.3 [7] If C is a binary self-dual code of length α andD is a quaternary self-dual code of lengthβ, thenC× Dis aZ2Z4-additive self-dual code of lengthα+β.
We say that a binary codeCisantipodalif, for any codewordz ∈C, we have thatz+1∈C.
IfC is aZ2Z4-additive code, we say thatC is antipodal if Φ(C)is antipodal. Clearly, aZ2Z4- additive codeC ⊆Zα2 ×Zβ4 is antipodal if and only if(1α,2β)∈ C.
We define the Type of aZ2Z4-additive self-dual codeC in terms of the weights of its code- words. If C has odd weights, then it is said to be Type 0. If it has only even weights, then the code is said to be Type I. If all the codewords have doubly-even weight, then it is said to be Type II. In general, if all the codewords of C have even weights, then C is an even code;
otherwise, C is an odd code. We remark that applying Theorem 4.3 to a binary self-dual code and a quaternary self-dual code gives a Type I code, and applying Theorem 4.3 to a binary doubly-even self-dual code and a quaternary doubly-even code gives a Type II code.
Now, we give some relations among Type, separability and antipodality.
Proposition 4.4 [7] LetC be aZ2Z4-additive self-dual code.
(i) C is antipodal if and only ifC is of Type I or Type II.
(ii) IfC is separable, thenC is antipodal.
(iii) IfC is of Type 0, thenC is non-separable and non-antipodal.
The following examples show the existence of all possible cases we have described.
Example 4.5 (Type 0) Let D = {(00|00),(00|22),(11|02),(11|20)}. Then, the code C1 = D ∪(D+ (01|11)) is a Z2Z4-additive self-dual code of type (2,2; 1,1; 1) and has generator matrix
G1 =
( 1 1 2 0 0 1 1 1
) .
The weight enumerator of this code isWC(x, y) = x6+ 4x3y3+ 3x2y4.Note that it has code- words of odd weight, hence it is a Type 0 code and by Proposition 4.4 it is non-separable.
Example 4.6 (Type I, separable) A Z2Z4-additive self-dual code with α, β ≥ 1 should have α ≥2, sinceαmust be even. AZ2Z4-additive self-dual code with minimum length hasα = 2, β = 1and2κ+β = 21+1 = 4codewords. For example,C2 = {(00|0),(00|2),(11|0),(11|2)}is aZ2Z4-additive self-dual code of type(2,1; 2,0; 1)and has generator matrix
G2 =
( 1 1 0 0 0 2
) .
Notice that forα = 2 andβ = 1, it is not possible to have odd weight codewords. Thus, the code must be of Type I and antipodal. Also, we have that the code restricted to the quaternary coordinates is{0,2}, which is self-dual and hence, by Theorem 4.2,C2is separable.
Example 4.7 (Type I, non-separable) The codes C3 andC4, generated byG3 andG4, respec- tively, are Type I non-separable, where
G3 =
1 1 1 1 0 0 0 0 0 1 0 1 2 0 0 0 0 1 0 1 0 2 0 0 0 1 0 1 0 0 2 0 0 0 1 1 1 1 1 1
G4 =
1 1 1 1 0 0 0 0 0 0 0 1 0 1 2 2 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 1 0 1 0 0 0 2 0 0 0 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 0 1
.
Example 4.8 (Type II, separable) Let C be the extended binary Hamming code of length 8 and letDbe the quaternary linear code generated by
2 2 0 0 2 0 2 0 1 1 1 1
.
Then, |D| = 2241 = 24, which is the correct size to be self-dual. Clearly, D is quaternary self-orthogonal and hence self-dual. On the other hand, C is a binary self-dual code. Since both codes have only doubly-even weights, we conclude thatC5 =C× Dis Type II separable.
Example 4.9 (Type II, non-separable) The codeC6generated byG6 is self-orthogonal, where
G6 =
1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 2 0 0 0 0 0 0 0 0 1 1 0 0 2 0 0 0 0 0 0 0 1 1 0 0 0 2 0 0 0 0 1 1 0 1 1 1 1 1 1
.
Since|C6| = 28, the code C6 is self-dual. Clearly it is non-separable, since (C6)X is not self- orthogonal. On the other hand, it can be checked that all weights are doubly-even.
4.1.2. ALLOWABLE ALPHA AND BETA VALUES
The following lemma is easily proven.
Lemma 4.10 [7] IfC is aZ2Z4-additive self-dual code of type(α, β;γ, δ;κ)andDis aZ2Z4- additive self-dual code of type(α′, β′;γ′, δ′;κ′), thenC × D is aZ2Z4-additive self-dual code of type(α+α′, β+β′;γ+γ′, δ+δ′;κ+κ′).
The following statements show some conditions for the parameters of aZ2Z4-additive self- dual code depending on its Type.
Theorem 4.11 [7] LetCbe aZ2Z4-additive self-dual code of type(α, β;γ, δ;κ), withα, β > 0.
(i) IfC is Type 0, thenα≥2,β ≥2.
(ii) IfC is Type I and separable, thenα≥2,β ≥1.
(iii) IfC is Type I and non-separable, thenα≥4,β ≥4.
(iv) IfC is Type II, thenα≥8,β ≥4.
Letαmin,βmin be the minimum values ofαandβgiven in Theorem 4.11 for each case (i) to(iv). Note that the codesC1,C2,C3,C5 andC6 are examples whereαandβare the minimum valuesαminandβmin.
Theorem 4.12 [7] Letαminandβminbe as defined above.
(i) There exist a Type 0 or Type I code of type(α, β;γ, δ;κ)if and only if α = αmin + 2a, a≥0,β ≥βmin.
(ii) There exist a Type II code of type (α, β;γ, δ;κ) if and only if α = αmin + 8a, β = βmin+ 4b,a, b≥0.
Finally, we remark a special case where the Gray map image is also a self-dual code.
Theorem 4.13 [7] IfC is a Type II code andΦ(C)is linear, thenΦ(C)is a binary doubly-even self-dual code.
4.1.3. WEIGHT ENUMERATORS
Let C be a Z2Z4-additive self-dual code of type (α, β;γ, δ;κ), and let C = Φ(C). Since the weight enumerator ofC satisfies (6), the codeC is held invariant by the action of the matrix
M = 1
√2
( 1 1 1 −2
)
, (10)
which satisfies M2 = I2. We also know that the length of C, n = α + 2β, is even, so WC(−x,−y) =WC(x, y)and so the weight enumerator is held invariant by the matrix
B =
( −1 0 0 −1
)
. (11)
Theorem 4.14 [7] LetC be aZ2Z4-additive self-dual code. Then,
WC(x, y)∈C[x2+y2, y(x−y)], ifC is Type 0, WC(x, y)∈C[x2+y2, x2y2(x2 −y2)2], ifC is Type I, WC(x, y)∈C[x8+ 14x4y4+y8, x4y4(x4−y4)4], ifC is Type II.
(12)
The results on the characterization of Z2Z4-additive self-dual codes of Type 0, Type I and Type II are summarized in the following table:
Type 0 Type I Type II
separability non-separable separable separable or non-separable or non-separable antipodality non-antipodal antipodal antipodal
separable - α= 2 + 2a α= 8 + 8a
α, β;a, b≥0 - β = 1 +b β = 4 + 4b
non-separable α= 2 + 2a α= 4 + 2a α= 8 + 8a α, β;a, b≥0 β = 2 +b β = 4 +b β = 4 + 4b
WC(x, y) C[x2+y2, C[x2 +y2, C[x8+ 14x4y4+y8, y(x−y)] x2y2(x2−y2)2] x4y4(x4−y4)4]
4.2. Construction technique ofZ2Z4-additive self-dual codes 4.2.1. USING THE SHADOW OF THE CODE
Consider the code given in Example 4.5. The codewords that have even weight are precisely {(00|00),(11|20),(11|02),(00|22)}. These form a linear subcode consisting of exactly half the codewords. This fact holds in general, that is, if C is a Type 0 code, then the subcode C0 = {v |v ∈ C, w(v) ≡ 0 (mod 2)}is a linear subcode with |C| = 2|C0|, andWC0(x, y) =
1
2(WC(x,−y) +WC(x, y)).Note that for binary linear self-dual codes of Type I, a similar prop- erty is satisfied except that C0 consists of doubly-even codewords. This notion cannot be ex- tended here toZ2Z4-additive self-dual codes of Type I, since the sum of two codewords with doubly-even weight may not have doubly-even weight.
We define the shadow of a Z2Z4-additive code C to be S = C0⊥\C. The shadow is a non- linear code with|S| = |C|.Recall that the matrixM given in (10) represents the action of the MacWilliams relations. Shadows of binary codes first appeared in [65] but were first specifically labeled as a code in [20]. The shadow has been generalized to numerous alphabets, see [53] for a complete description. For a Z2Z4-additive self-dual code of Type 0, the weight enumerator of the shadowS isWS(x, y) = |C|1 M ·WC(x,−y) =WC
(x+y√
2,−(x√−y) 2
)
[7].The difference with the usual binary case is that these weight enumerators are not necessarily possible weight enu- merators for binary self-dual codes, since in the case ofZ2Z4-additive codes of Type 0 it may be odd weight vectors. Given a possible weight enumerator for Type 0 codes, one can compute the weight enumerator of the shadow.
Example 4.15 LetCbe theZ2Z4-additive self-dual code given in Example 4.5. We can compute WC(x, y) = x6+ 4x3y3+ 3x2y4,
WC0(x, y) = x6+ 3x2y4,
WS(x, y) = 3x4y2+ 4x3y3+y6.
Note thatC0 =D, and the shadow isS=C0⊥\C ={(11|00),(01|11),(10|13),(00|20),(01|33), (00|02),(11|22),(10|31)}.
The codeC0 has 4 cosets inC0⊥. LetC0,0 =C0 andC1,0 =C\C0,0.LetC0⊥ =C ∪ C0,1∪ C1,1, i.e. S = C0,1 ∪ C1,1.There are vectors s andt such that C = ⟨C0, t⟩ andC0⊥ = ⟨C, s⟩. Then, we have that Ci,j = C0 +it +js. Taking s = (1α,2β), C0,0 and C0,1 consist of even weight vectors andC1,0 andC1,1 consist of odd weight vectors. Hence we have the orthogonality given in Table 1.
C0,0 C1,0 C0,1 C1,1
C0,0 0 0 0 0
C1,0 0 0 2 2
C0,1 0 2 0 2
C1,1 0 2 2 0
Table 1: Orthogonality Relations
Proposition 4.16 [7] LetC be a Type 0 code. Then, the codesC0,0 ∪ C0,1 = ⟨C, s⟩andC0,0∪ C1,1 =⟨C, s+t⟩are self-dual codes that are not Type 0.
We can now generalize the construction first described in [17] but greatly expanded in [28].
Theorem 4.17 [7] LetC andD be Type 0 codes in Zα2 ×Zβ4 andZα2′ ×Zβ4′, respectively. If
⟨Ci,j, Ci,j⟩=⟨Di,j, Di,j⟩, for alli, j ∈ {0,1}, then the code
(C0,0,D0,0)∪(C0,1,D0,1)∪(C1,0,D1,0)∪(C1,1,D1,1)
is a self-dual code inZα+α2 ′ ×Zβ+β4 ′.If⟨Ci,j, Ci,j⟩ ̸=⟨Di,j, Di,j⟩, for somei, j ∈ {0,1}, then the code
(C0,0,D0,0)∪(C0,1,D1,1)∪(C1,0,D1,0)∪(C1,1,D0,1) is a self-dual code inZα+α2 ′ ×Zβ+β4 ′.
4.2.2. EXTENDING THE LENGTH
Let C be a self-dual code in Zα2 × Zβ4 and let v ∈ Zα2 × Zβ4 such that v /∈ C. We define Cv ={u∈ C | ⟨u, v⟩= 0}.It is immediate thatCv is a subgroup ofC and that the index[C :Cv] is either 2 or 4. In either case, we have that[C :Cv] = [Cv⊥ :C]and thatCv⊥ = ⟨C, v⟩.Letwbe the vector such thatC =⟨Cv, w⟩.We can then writeCv⊥=⟨Cv, w, v⟩.
Example 4.18 LetC1 be theZ2Z4-additive self-dual code given in Example 4.5, generated by G1, and letv = (00|20). Then,Cv ={(00|00),(11,|20),(00|22),(11|02)}, generated by
Gv =
( 1 1 2 0 0 0 2 2
) ,
andC =⟨Cv, w⟩, wherew= (01|11). Therefore, the codeCv⊥is generated by
Hv =
1 1 2 0 0 0 2 2 0 0 2 0 0 1 1 1
.
We can form a codeC¯by extending the codeC =Cv⊥in the following manner. Foru∈ Cv⊥
we letu¯= (u′X, uX, uY, u′Y), whereu′X is an extension of the binary part andu′Y is an extension of the quaternary part. Then letC¯ = ⟨{u¯ |u ∈ Cv⊥}⟩. We choose u′X and u′Y so that C¯ is a self-orthogonal code. We denote byα′ the length ofu′X and byβ′ the length ofu′Y.IfC¯ is not self-dual, then we may need to add additional vectors to the code. In all cases, we letu′X and u′Y be0 if u ∈ Cv, and we denote byC¯v the extension of Cv. SinceC = ⟨Cv, w⟩, we denote C¯=⟨C¯v,w¯⟩. We separate the construction into three cases, namely Case 1 is whenβ′ = 0, Case 2 is whenα′ = 0, and Case 3 is when neither are 0.
Theorem 4.19 [7] LetCbe aZ2Z4-additive code of type(α, β;γ, δ;κ)andv ̸∈ C. Letw,Cvbe as before andC = Cv⊥ = ⟨Cv, w, v⟩. There exists aZ2Z4-additive self-dual codeD =⟨C, V¯ ⟩ of type(α+α′, β+β′;γ′, δ′;κ′), for some set of vectorsV with the following conditions:
(i) α′ ̸= 0andβ′ = 0only if⟨v, w⟩= 2and⟨v, v⟩ ∈ {0,2},
(ii) α′ = 0andβ′ ̸= 0only if⟨v, w⟩= 2or⟨v, w⟩ ∈ {1,3}and⟨v, v⟩ ∈ {1,3}, (iii) α′ ̸= 0andβ′ ̸= 0.
The steps to obtain an extendedZ2Z4-additive codeC¯from aZ2Z4-additive self-dual code C are the following. First, selectv ̸∈ C,w∈ C\Cv such thatC =⟨Cv, w⟩, with the conditions in
⟨v, v⟩,⟨v, w⟩described in Theorem 4.19. After that, determine the values ofvX′ , vY′ , w′X, wY′ , V from Tables 2-6. Finally, ifGv is the generator matrix ofCv, then the generator matrix ofC¯is
GD =
0 Gv 0 v′X v vY′ w′X w wY′
V
.
⟨v, v⟩ v′X wX′ V
0 (0,0,1,1) (0,1,0,1) {(1,1,1,1,0)}
2 (0,1) (1,1) ∅
Table 2: Caseα′ ̸= 0, β′ = 0.
⟨v, v⟩ vY′ w′Y V
0 (1,1,1,1) (2,0,0,0) {(0,0,2,2,0),(0,0,0,2,2)} 1 (1,1,1) (2,0,0) {(0,0,2,2)}
2 (1,1) (2,0) ∅
3 (1) (2) ∅
Table 3: Caseα′ = 0, β′ ̸= 0,⟨v, w⟩= 2.
⟨v, v⟩ vY′ w′Y V
1 (1,1,1,0) (1,1,1,1) {(0,0,2,2,0),(0,2,2,0,0)} 3 (3,0,0,0) (1,1,1,1) {(0,0,2,2,0),(0,0,0,2,2)}
Table 4: Caseα′ = 0, β′ ̸= 0,⟨v, w⟩= 1.
⟨v, v⟩ vX′ vY′ w′X w′Y V
0 (1,0) (1,0,1) (1,0) (1,1,0) {(1,1,0,2,0,0),(1,1,0,0,2,0)} 1 (1,0) (1,0) (1,0) (1,1) {(1,1,0,2,0)}
2 (1,1) (0,1,1) (1,0) (1,1,0) {(1,1,0,2,0,0),(1,1,0,0,2,2)} 3 (1,1) (1,0) (1,0) (1,1) {(1,1,0,0,2)}
Table 5: Caseα′ ̸= 0, β′ ̸= 0,⟨v, w⟩= 1.
⟨v, v⟩ vX′ vY′ wX′ wY′ V
0 (1,0) (1,1) (1,1) (2,2) {(1,1,0,2,0)} 1 (1,0) (1,0) (1,1) (0,2) {(1,1,0,2,0)} 2 (1,1) (1,3) (1,0) (1,1) ∅ 3 (0,0) (0,1) (1,1) (0,2) {(1,1,0,2,0)}
Table 6: Caseα′ ̸= 0, β′ ̸= 0,⟨v, w⟩= 2.
4.2.3. NEIGHBOR CONSTRUCTION
Let C be a self-dual code in Zα2 ×Zβ4 and let v be a self-orthogonal vector such that v ̸∈ C. As before, we denote by Cv the subcode Cv = {u ∈ C | ⟨u, v⟩ = 0}. LetN(C, v) = ⟨Cv, v⟩. The following construction technique is a generalization of the technique used for the neighbor construction for codes over finite fields.
Theorem 4.20 [7] LetCbe aZ2Z4-additive self-dual code and letvbe a self-orthogonal vector such thatv ̸∈ C. Then,N(C, v)is a self-dual code.
Theorem 4.21 [7] EveryZ2Z4-additive self-dual code can be found by repeated application of the neighbor code from any self-dual code of that length.
4.3. Z2Z4-additive formally self-dual codes
In general, a codeCover any ring is said to beformally self-dualif its weight enumerator is the same as the weight enumerator of its orthogonal. For example, any self-dual code is necessarily formally self-dual but, of course, there are formally self-dual codes that are not self-dual. For quaternary codes, a code can be formally self-dual with respect to the Lee weight enumerator but not with respect to the Hamming weight enumerator and vice versa.
AZ2Z4-additive codeC isformally self-dualifWC⊥(x, y) = WC(x, y), with respect to the weight enumerator given in (5). In this subsection, we summarizes the results given in [25].
Example 4.22 LetC andC be the codes generated by the following matrices ( 0 1 0
1 0 0 )
, (
0 0 1 ) ,
respectively. It is clear that C⊥ = D and that the weight enumerator of both is WC(x, y) = WC⊥(x, y) =x4+ 2x3y+x2y2. Hence, these codes areZ2Z4-additive formally self-dual. The
