Z4-linear Codes Research Articles

On generalized quasi-cyclic codes over [formula omitted

Based on good algebraic structures and practicabilities, generalized quasi-cyclic (GQC) codes play important roles in coding theory. In this paper, we study some results on GQC codes over Z4 including normalized generating sets, minimal generating sets and normalized generating sets of their dual codes. As an application, new Z4-linear codes and good binary nonlinear codes are constructed from GQC codes over Z4.

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.

SISO Decoding of ℤ4 Linear Kerdock and Preparata Codes

Some nonlinear codes, such as Kerdock and Preparata codes, can be represented as binary images under the Gray map of linear codes over rings. This paper introduces MAP decoding of Kerdock and Preparata codes by working with their quaternary representation (linear codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{4}$ </tex-math></inline-formula> ) with the complexity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(N^{2}\log _{2} N)$ </tex-math></inline-formula> , where N is the code length in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{4}$ </tex-math></inline-formula> . A sub-optimal bitwise APP decoder with good error-correcting performance and complexity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(N\log _{2} N)$ </tex-math></inline-formula> that is constructed using the decoder lifting technique is also introduced. This APP decoder extends upon the original lifting decoder by working with likelihoods instead of hard decisions and is not limited to Kerdock and Preparata code families. Simulations show that our novel decoders significantly outperform several popular decoders in terms of error rate.

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.

Propelinear codes related to some classes of optimal codes

A code is said to be propelinear if its automorphism group contains a subgroup that acts regularly on codewords. We show propelinearity of complements of cyclic codes C 1,i , (i, 2 m − 1) = 1, of length n = 2 m − 1, including the primitive two-error-correcting BCH code, to the Hamming code; the Preparata code to the Hamming code; the Goethals code to the Preparata code; and the Z4-linear Preparata code to the Z4-linear perfect code.

Comparing decoding methods for quaternary linear codes

Permutation decoding is a technique which involves finding a subset S, called PD-set, of the permutation automorphism group of a code C. Constructions of small PD-sets for partial decoding for two families of Z4-linear codes (Hadamard and Kerdock) are given. Moreover, different decoding methods for Z4-linear codes are compared by showing their performance applied to these two families.

Some classes of linear codes over $$\mathbb {Z}_4+v\mathbb {Z}_4$$ Z 4 + v Z 4 and their applications to construct good and new $$\mathbb {Z}_4$$ Z 4 -linear codes

Some classes of linear codes over the ring $$\mathbb {Z}_4+v\mathbb {Z}_4$$Z4+vZ4 with $$v^2=v$$v2=v are considered. Construction of Euclidean formally self-dual codes and unimodular complex lattices from self-dual codes over $$\mathbb {Z}_4+v\mathbb {Z}_4$$Z4+vZ4 are studied. Structural properties of cyclic codes and quadratic residue codes are also considered. Finally, some good and new $$\mathbb {Z}_4$$Z4-linear codes are constructed from linear codes over $$\mathbb {Z}_4+v\mathbb {Z}_4$$Z4+vZ4.

Cyclic and some constacyclic codes over the ring [formula omitted

In this paper, we study cyclic codes and constacyclic codes with shift constant (2+u) over R=Z4+uZ4, where u2=1. We determine the form of the generators of the cyclic codes over this ring and their spanning sets. Considering their Z4 images, we prove that the Z4-image of a (2+u)-constacyclic code of odd length is a cyclic code over Z4. We also present many examples of cyclic codes over R whose Z4-images have better parameters than previously best-known Z4-linear codes.

Families of Hadamard &lt;formula formulatype="inline"&gt; &lt;tex Notation="TeX"&gt;$\BBZ_{2}\BBZ_{4}Q_{8}$&lt;/tex&gt;&lt;/formula&gt;-Codes

A Z2Z4Q8-code is the binary image, after a Gray map, of a subgroup of Z2k1 × Z4k2 × Q8k3, where Q8 is the quaternion group on eight elements. Such Z2Z4Q8-codes are translation invariant propelinear codes as are the well known Z4-linear or Z2Z4-linear codes. In this paper, we show that there exist “pure” Z2Z4Q8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the Z2Z4Q8-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of Z2Z4Q8-codes; we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard.

Minimum Weights and Weight Enumerators of $\BBZ_{4}$-Linear Quadratic Residue Codes

A fast method to compute the minimum Lee weight and the symmetrized weight enumerator of extended quadratic residue codes (XQR-codes) over the ring Z4 is developed. Our approach is based on the classical Brouwer-Zimmermann algorithm and additionally takes advantage of the large group of automorphisms and the self-duality of the Z4-linear XQR-codes as well as the projection to the binary XQR-codes. As a result, the hitherto unknown minimum Lee distances of all Z4-linear XQR-codes of lengths between 72 and 104 and the minimum Euclidean distances for the lengths 72, 80, and 104 are computed. It turns out that the binary Gray image of the Z4-linear XQR-codes of lengths 80 and 104 has higher minimum distance than any known linear binary code of equal length and cardinality. Furthermore, the Z4-linear XQR-code of length 80 is a new example of an extremal Z4-linear typeII code. Additionally, we give the symmetrized weight enumerator of the Z4-linear XQR-codes of lengths 72 and 80, and we correct the weight enumerators of the Z4-linear XQR-code of length 48 given by Pless and Qian and Bonnecaze et al.

Construction of ${\BBZ}_4$-Linear Reed–Muller Codes

In this paper, new quaternary Plotkin constructions are given 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 (RM) codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters and fundamental properties as the codes in the usual binary linear RM family. To make the duality relationships in the constructed families more evident, the concept of Kronecker inner product is introduced.

Bent functions with stronger nonlinear properties: K-bent functions

We introduce the notion of k-bent function, i.e., a Boolean functionwith evennumber m of variables υ1, …, υm which can be approximated with all functions of the form 〈u, v〉j ⊕ a in the equally poor manner, where u ∈ ℤ2m, a ∈ ℤ2, and 1 ⩽ j ⩽ k. Here 〈·, ·〉j is an analog of the inner product of vectors; k changes from 1 to m/2. The operations 〈·, ·〉k, 1 ⩽ k ⩽ m/2, are defined by using the special series of binary Hadamard-like codes Amk of length 2m. Namely, the vectors of values for the functions 〈u, v〉k ⊕ a are codewords of the code Amk. The codes Amk are constructed using subcodes of the ℤ4-linear Hadamard-like codes of length 2m+1, which were classified by D. S. Krotov (2001). At that the code Am1 is linear and Am1, …, Amm/2 are pairwise nonequivalent. On the codewords of a code Amk, we define a group operation • coordinated with the Hamming metric. That is why we can consider k-bent functions as functions that are maximal nonlinear in k distinct senses of linearity at the same time. Bent functions in usual sense coincide with 1-bent functions. For k > l ⩾ 1, the class of k-bent functions is a proper subclass of the class of l-bent functions. In the paper, we give methods for constructing k-bent functions and study their properties. It is shown that there exist k-bent functions with an arbitrary algebraic degree d, where 2 ⩽ d ⩽ max {2, m/2 − k + 1}. Given an arbitrary k, we define the subset \( \mathfrak{F}_m^k \mathcal{F}_m^k \) of the set \( \mathfrak{F}_m^k \mathcal{F}_m^k \)\( \mathcal{A}_m^k \mathcal{B}_m^k \) of all Boolean functions in m variables with the following property: on this subset k-bent functions and 1-bent functions coincide.

Symplectic semifield planes and ℤ₄–linear codes

There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and Z 4 \mathbb {Z}_4 –linear Kerdock and Preparata codes on the other. These inter–relationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of non–isotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of Z 4 \mathbb {Z}_4 –linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2. We also obtain large numbers of “boring” affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes “boring” in the sense that each has as small an automorphism group as possible. The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.

Z4 -Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line-Sets

When m is odd, spreads in an orthogonal vector space of type Ω+(2m + 2,2) are related to binary Kerdock codes and extremal line-sets in 2m + 1 with prescribed angles. Spreads in a 2m-dimensional binary symplectic vector space are related to Kerdock codes over Z4 and extremal line-sets in \CC 2 m with prescribed angles. These connections involve binary, real and complex geometry associated with extraspecial 2-groups. A geometric map from symplectic to orthogonal spreads is shown to induce the Gray map from a corresponding Z4-Kerdock code to its binary image. These geometric considerations lead to the construction, for any odd composite m, of large numbers of Z4-Kerdock codes. They also produce new Z4-linear Kerdock and Preparata codes. 1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.