Gray Map Image Research Articles

Partial permutation decoding and PD-sets for [formula omitted]-linear generalized Hadamard codes

It is known that Zps-linear codes, which are the Gray map image of Zps-additive codes (linear codes over Zps), are systematic and a systematic encoding has been found. This makes Zps-linear codes suitable to apply the permutation decoding method. This technique is also based on the existence of r-PD-sets, which are subsets of the permutation automorphism group of the code. In this paper, we study the permutation automorphism group of Zps-linear generalized Hadamard codes of type (n;t1,…,ts) and show how to construct r-PD-sets of size r+1, for all r up to an upper bound, in order to be able to perform a partial permutation decoding for these codes.

Linearity and classification of [formula omitted]-linear generalized Hadamard codes

The ZpZp2-additive codes are subgroups of Zpα1×Zp2α2, and can be seen as linear codes over Zp when α2=0, Zp2-additive codes when α1=0, or Z2Z4-additive codes when p=2. A ZpZp2-linear generalized Hadamard (GH) code is a GH code over Zp which is the Gray map image of a ZpZp2-additive code. Recursive constructions of ZpZp2-additive GH codes of type (α1,α2;t1,t2) with t1,t2≥1 are known. In this paper, we generalize some known results for ZpZp2-linear GH codes with p=2 to any p≥3 prime when α1≠0, and then we compare them with the ones obtained when α1=0. First, we show for which types the corresponding ZpZp2-linear GH codes are nonlinear over Zp. Then, for these codes, we compute the kernel and its dimension, which allow us to classify them completely. Moreover, by computing the rank of some of these codes, we show that, unlike Z4-linear Hadamard codes, the Zp2-linear GH codes are not included in the family of ZpZp2-linear GH codes with α1≠0 when p≥3 prime. Indeed, there are some families with infinite nonlinear ZpZp2-linear GH codes, where the codes are not equivalent to any Zps-linear GH code with s≥2.

On the constructions of [formula omitted]-linear generalized Hadamard codes

The ZpZp2-additive codes are subgroups of Zpα1×Zp2α2, and can be seen as linear codes over Zp when α2=0, Zp2-additive codes when α1=0, or Z2Z4-additive codes when p=2. A ZpZp2-linear generalized Hadamard (GH) code is a GH code over Zp which is the Gray map image of a ZpZp2-additive code. In this paper, we generalize some known results for ZpZp2-linear GH codes with p=2 to any p≥3 prime when α1≠0. First, we give a recursive construction of ZpZp2-additive GH codes of type (α1,α2;t1,t2) with t1,t2≥1. We also present many different recursive constructions of ZpZp2-additive GH codes having the same type, and show that we obtain permutation equivalent codes after applying the Gray map. Finally, according to some computational results, we see that, unlike Z4-linear GH codes, when p≥3 prime, the Zp2-linear GH codes are not included in the family of ZpZp2-linear GH codes with α1≠0. Indeed, we observe that the constructed codes are not equivalent to the Zps-linear GH codes for any s≥2.

On $\mathbb{Z}_{\text{8}}$ -Linear Hadamard Codes: Rank and Classification

The $\mathbb {Z}_{2^{s}}$ -additive codes are subgroups of $\mathbb {Z}^{n}_{2^{s}}$ , and can be seen as a generalization of linear codes over $\mathbb {Z}_{2}$ and $\mathbb {Z}_{4}$ . A $\mathbb {Z}_{2^{s}}$ -linear Hadamard code is a binary Hadamard code which is the Gray map image of a $\mathbb {Z}_{2^{s}}$ -additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the $\mathbb {Z}_{4}$ -linear Hadamard codes. However, when $s > 2$ , the dimension of the kernel of $\mathbb {Z}_{2^{s}}$ -linear Hadamard codes of length $2^{t}$ only provides a complete classification for some values of $t$ and $s$ . In this paper, the rank of these codes is computed for $s=3$ . Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once $t\geq 3$ is fixed. In this case, the number of nonequivalent such codes is also established.

Equivalences among [formula omitted]-linear Hadamard codes

The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z2s-linear Hadamard codes of length 2t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t=11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s∈{2,3}, the full classification of the Z2s-linear Hadamard codes of length 2t is established by giving the exact number of such codes.

Partial Permutation Decoding for Several Families of Linear and <inline-formula> <tex-math notation="LaTeX">${\mathbb{Z}}_4$ </tex-math> </inline-formula>-Linear Codes

A general criterion to obtain $s$ -PD-sets of minimum size $s+1$ for partial permutation decoding, which enable correction up to $s$ errors, for systematic codes over a finite field ${ {F}}_{q}$ and ${ {Z}}_{4}$ -linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over ${ {F}}_{q}$ , ${ {Z}}_{4}$ -linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new $s$ -PD-sets of size $s+1$ for ${ {Z}}_{4}$ -linear Hadamard codes of type $2^\gamma 4^\delta $ , for all $\delta \geq 4$ and $1 ; and for ${ {Z}}_{4}$ -linear simplex codes of type $4^{m}$ , for all $m\geq 2$ and $1 , are also provided.

On the Rank of [formula omitted]-linear Hadamard Codes

The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z4-linear Hadamard codes. However, when s>2, the dimension of the kernel of Z2s-linear Hadamard codes of length 2t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is given for s=3. Moreover, it is shown that this invariant, along with the dimension of the kernel, provides a complete classification, once t≥3 is fixed. In this case, the number of nonequivalent such codes is also established.

On $$\mathbb {Z}_{2^s}$$ Z 2 s -linear Hadamard codes: kernel and partial classification

The $$\mathbb {Z}_{2^s}$$ -additive codes are subgroups of $$\mathbb {Z}^n_{2^s}$$ , and can be seen as a generalization of linear codes over $$\mathbb {Z}_2$$ and $$\mathbb {Z}_4$$ . A $$\mathbb {Z}_{2^s}$$ -linear Hadamard code is a binary Hadamard code which is the Gray map image of a $$\mathbb {Z}_{2^s}$$ -additive code. It is known that the dimension of the kernel can be used to give a complete classification of the $$\mathbb {Z}_4$$ -linear Hadamard codes. In this paper, the kernel of $$\mathbb {Z}_{2^s}$$ -linear Hadamard codes of length $$2^t$$ and its dimension are established for $$s > 2$$ . Moreover, we prove that this invariant only provides a complete classification for some values of t and s. The exact amount of nonequivalent such codes are given up to $$t=11$$ for any $$s\ge 2$$ , by using also the rank.

New results on quaternary codes and their Gray map images for constructing uniform designs

The research of developing efficient methodologies for constructing optimal experimental designs has been very active in the last decade. Uniform design is one of the most popular approaches, carried out by filling up experimental points in a determinately uniform fashion. Applications of coding theory in experimental design are interesting and promising. Quaternary codes and their binary Gray map images attracted much attention from those researching design of experiments in recent years. The present paper aims at exploring new results for constructing uniform designs based on quaternary codes and their binary Gray map images. This paper studies the optimality of quaternary designs and their two and three binary Gray map image designs in terms of the uniformity criteria measured by: the Lee, wrap-around, symmetric, centered and mixture discrepancies. Strong relationships between quaternary designs and their two and three binary Gray map image designs are obtained, which can be used for efficiently constructing two-level designs from four-level designs and vice versa. The significance of this work is evaluated by comparing our results to the existing literature.

Partial permutation decoding for binary linear and $$Z_4$$ Z 4 -linear Hadamard codes

In this paper, s- $${\text {PD}}$$ -sets of minimum size $$s+1$$ for partial permutation decoding for the binary linear Hadamard code $$H_m$$ of length $$2^m$$ , for all $$m\ge 4$$ and $$2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1$$ , are constructed. Moreover, recursive constructions to obtain s- $${\text {PD}}$$ -sets of size $$l\ge s+1$$ for $$H_{m+1}$$ of length $$2^{m+1}$$ , from an s- $${\text {PD}}$$ -set of the same size for $$H_m$$ , are also described. These results are generalized to find s- $${\text {PD}}$$ -sets for the $${\mathbb {Z}}_4$$ -linear Hadamard codes $$H_{\gamma , \delta }$$ of length $$2^m$$ , $$m=\gamma +2\delta -1$$ , which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type $$2^\gamma 4^\delta $$ . Specifically, s-PD-sets of minimum size $$s+1$$ for $$H_{\gamma , \delta }$$ , for all $$\delta \ge 3$$ and $$2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1$$ , are constructed and recursive constructions are described.

Construction and classification of [formula omitted]-linear Hadamard codes

The Z2s-additive and Z2Z4-additive codes are subgroups of Z2sn and Z2α×Z4β, respectively. Both families can be seen as generalizations of linear codes over Z2 and Z4. A Z2s-linear (resp. Z2Z4-linear) Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive (resp. Z2Z4-additive) code. It is known that there are exactly ⌊t−12⌋ and ⌊t2⌋ nonequivalent Z2Z4-linear Hadamard codes of length 2t, with α=0 and α≠0, respectively, for all t≥3. In this paper, new Z2s-linear Hadamard codes are constructed for s>2, which are not equivalent to any Z2Z4-linear Hadamard code. Moreover, for each s>2, it is claimed that the new constructed nonlinear Z2s-linear Hadamard codes of length 2t are pairwise nonequivalent.

$$\mathbb {Z}_{4}$$ Z 4 -codes and their Gray map images as orthogonal arrays

A classic result of Delsarte connects the strength (as orthogonal array) of a linear code with the minimum weight of its dual: the former is one less than the latter. We show that Delsarte's observation extends to codes over arbitrary finite rings. Since the paper of Hammons \emph{et al.}, there is a lot of interest in codes over rings, especially in codes over $Z_4$ and their (usually non-linear) binary Gray map images. We show that Delsarte's observation extends to codes over arbitrary finite commutative rings with identity. Also, we show that the strength of the Gray map image of a $Z_4$ code is one less than the minimum Lee weight of its Gray map image.

Classification of the <inline-formula> <tex-math notation="LaTeX">$Z_{2}Z_{4}$ </tex-math></inline-formula>-Linear Hadamard Codes and Their Automorphism Groups

A $Z_2Z_4$-linear Hadamard code of length $\alpha+2\beta=2^t$ is a binary Hadamard code which is the Gray map image of a $Z_2Z_4$-additive code with $\alpha$ binary coordinates and $\beta$ quaternary coordinates. It is known that there are exactly $[(t-1)/2]$ and $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$, with $\alpha=0$ and $\alpha\not=0$, respectively, for all $t\geq 3$. In this paper, it is shown that each $Z_2Z_4$-linear Hadamard code with $\alpha=0$ is equivalent to a $Z_2Z_4$-linear Hadamard code with $\alpha\not=0$; so there are only $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$. Moreover, the order of the monomial automorphism group for the $Z_2Z_4$-additive Hadamard codes and the permutation automorphism group of the corresponding $Z_2Z_4$-linear Hadamard codes are given.

A Class of Constacyclic Codes over <i>R</i> + <i>vR</i> and Its Gray Image

We study (1 + 2v)-constacyclic codes overR + vR and their Gray images, where v2 + v = 0 and R is a finite chain ring with maximal ideal λ> and nilpotency index e. It is proved that the Gray map images of a (1 + 2v)-constacyclic codes of length n over R + vR are distance-invariant linear cyclic codes of length 2n over R. The generator polynomials of this kind of codes for length n are determined, where n is relatively prime to p, p is the character of the field R/λ> . Their dual codes are also discussed.

Quaternary Reed–Muller Codes

The class of quaternary Reed-Muller codes is introduced as a generalization of Reed-Muller codes to /spl Zopf//sub 4/-linear codes. Properties of this class of codes are established including the rank and kernel of the Gray map image of codes in this class. The class includes all /spl Zopf//sub 4/-linear Kerdock-like and Preparata-like codes.

Some constacyclic codes over Z2 k and binary quasi-cyclic codes

The concept of negacyclic code was recently introduced in Wolfmann (IEEE Trans. Inform. Theory 45 (1999) 2527–2532), in which some relations between the negacyclic codes and their Gray map images are proved. In this note, for k⩾1 an isometry ϕ k between codes over Z 2 k+1 and codes over Z 4 is introduced and used to give a generalization of the Gray map equivalent to the one given in Carlet (IEEE Trans. Inform. Theory 44 (1998) 1543–1547). Furthermore, by means of this isometry, the concept of negacyclic codes is extended to codes over the ring Z 2 k+1 , obtaining a class of constacyclic codes referred to as hpo-cyclic codes (half plus one-cyclic codes). A characterization of these codes in terms of their images under ϕ k is given. It is also proved that the generalized Gray map image of an hpo-cyclic code is a binary distance invariant (not necessarily linear) quasi-cyclic code. Finally, some linear hpo-cyclic codes are discussed and a few examples are given.

A Generalization of Negacyclic Codes

Abstract In this paper, for k ≥ 1 an isometry φk between codes over Z 2k+1 and codes over Z 4 is introduced and is used to give a generalization of the Gray map. Furthermore, by means of this isometry, the concept of negacyclic codes is extended to codes over the ring Z 2k+1, referred to as hpo-cyclic codes (half plus one-cyclic codes). A characterization of these codes in terms of their images under φk is given. It is also shown that the generalized Gray map image of a hpo-cyclic code is a binary distance invariant (not necessary linear) quasi-cyclic code. Finally, some linear hpo-cyclic codes are discussed.