Type-II polyadic constacyclic codes over finite fields

Let p be a prime number and m be a positive integer. In this paper, we investigate cyclic codes of length n over the local non-Frobenius ring R=GR(p2,m)[u], where u2=0 and pu=0. We first determine the algebraic structure of cyclic codes of arbitrary length n. For the case gcd(n,p)=1, we explicitly describe the generators of cyclic codes over R. Moreover, we establish necessary and sufficient conditions for the existence of self-dual and LCD codes, together with their enumeration. Several illustrative examples and tables are presented, highlighting the mass formula for cyclic self-orthogonal codes, cyclic LCD codes, and families of new cyclic codes that arise from our results.

Constacyclic codes are well-known generalizations of cyclic and negacyclic codes. Due to their rich algebraic structure, constacyclic codes are used to construct quantum codes and symbol-pair codes. Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> be a finite field with order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is a positive power of a prime <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> . Suppose that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is a positive integer and the product of distinct prime factors of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> divides <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q-1$ </tex-math></inline-formula> , i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$rad(n)\mid (q-1)$ </tex-math></inline-formula> . In this paper, we explicitly factorize the polynomial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{n}-\lambda $ </tex-math></inline-formula> for each <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda \in {\mathbb {F}}_{q}^{*}$ </tex-math></inline-formula> . As applications, first, we obtain all repeated-root <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> -constacyclic codes and their dual codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$np^{s}$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> ; second, we determine all simple-root LCD cyclic codes and LCD negacyclic codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> ; third, we list all self-dual repeated-root negacyclic codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$np^{s}$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> . In contrast to known results, the lengths of constacyclic codes in this paper have more flexible parameters.

Let $\mathbb {F}_{p^{m}}$ be a finite field with pm elements for an odd prime p and a positive integer m. In this paper, we aim to construct non-binary quantum codes from repeated root cyclic and negacyclic codes of length 4ps over $\mathbb {F}_{p^{m}}$ . To achieve this, first we discuss the structure of all maximum distance separable (MDS) cyclic and negacyclic codes of length 4ps over $\mathbb {F}_{p^{m}}$ and their Euclidean and Hermitian duals. Further, we establish a necessary and sufficient condition for these codes to contain their duals. Finally, many new and quantum MDS codes are obtained with the help of CSS and Hermitian constructions from dual containing cyclic and negacyclic codes.

Cyclic codes are important in theory, as they are closely related to a number of areas of mathematics. Cyclic codes are also important in practice, as they have efficient encoding and decoding algorithms. An infinite family of cyclic codes over GF( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ) is said to have linearly-best-known parameters if for any [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ] code <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> in this family, there is no known [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ′] linear code over GF( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ) such that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ′ > <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> . An infinite family of cyclic codes over GF( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ) is said to have cyclicly-best-known parameters if for any [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ] code <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> in this family, there is no known [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ′] cyclic code over GF( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ) such that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> ′ > <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> . It is very rare to see an infinite family of binary cyclic codes with cyclicly-best-known parameters whose duals codes have also cyclicly-best-known parameters. The objective of this paper is to study such family of binary cyclic codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>m</i></sup> – 1 and dimension 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>m</i></sup> – 1 – <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> – 1)/2, denoted by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2,<i>m</i>,2)</sub> , and their dual codes <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2,<i>m</i>,2)</sub> . The weight distribution of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2,<i>m</i>,2)</sub> is settled and the parameters of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2,<i>m</i>,2)</sub> are investigated in this paper. A larger family of binary cyclic codes <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2,<i>m</i>,<i>r</i>)</sub> and their duals are also constructed and studied in this paper, where 0 ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> – 1.

&lt;abstract&gt;&lt;p&gt;In this paper, we propose a way to partition any constacyclic code over a finite field in its equivalence classes according to the algebraic structure of the code. Such a method gives the generalization of cyclically permutable codes (CPCs), which are called constacyclically permutable codes (CCPCs), and it is useful to derive a CCPC from a given constacyclic code. Moreover, we present an enumerative formula for the code size of such a CCPC, with all of the terms being positive integers, and we provide an algebraic method to produce such a CCPC.&lt;/p&gt;&lt;/abstract&gt;

The finite field $\mathbb {F}_{q^\ell }$ of $q^\ell $ elements contains $\mathbb {F}_{q}$ as a subfield. If $\theta \in \mathbb {F}_{q^\ell }$ is of degree $\ell $ over $\mathbb {F}_{q}$ , it can be used to unfold elements of $\mathbb {F}_{q^\ell }$ to vectors in $\mathbb {F}_{q}^\ell $ . We apply the unfolding to the coordinates of all codewords of a cyclic code $\mathcal {C}$ over $\mathbb {F}_{q^\ell }$ of length $n$ . This generates a quasi-cyclic code $\mathcal {Q}$ over $\mathbb {F}_{q}$ of length $n\ell $ and index $\ell $ . We focus on the class of quasi-cyclic codes resulting from the unfolding of cyclic codes. Given a generator polynomial $g(x)$ of a cyclic code $\mathcal {C}$ , we present a formula for a generator polynomial matrix for the unfolded code $\mathcal {Q}$ . On the other hand, for any quasi-cyclic code $\mathcal {Q}$ with a reduced generator polynomial matrix $G$ , we provide a necessary and sufficient condition on $G$ that determines whether or not the code $\mathcal {Q}$ can be represented as the unfolding of a cyclic code. Furthermore, as an application, we discuss the reversibility of the class of quasi-cyclic codes resulting from unfolding of cyclic codes. Specifically, we provide a necessary and sufficient condition on the defining set $\mathcal {T}$ of the cyclic code $\mathcal {C}$ that ensures the reversibility of the unfolded code. Numerical examples are used to illustrate theoretical results. Some of these examples show that quasi-cyclic codes reversibility does not necessarily require a self-reciprocal generator polynomial for the cyclic code. Since reversibility is essential in constructing DNA codes, some DNA codes are designed as examples.

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. It was proved that asymptotically good Hermitian LCD codes exist. The objective of this paper is to construct some cyclic Hermitian LCD codes over finite fields and analyse their parameters. The dimensions of these codes are settled and the lower bounds on their minimum distances are presented. Most Hermitian LCD codes presented in this paper are not BCH codes.

Locally recoverable (LRC) codes and their variants have been extensively studied in recent years. In this paper we focus on cyclic constructions of LRC codes and derive conditions on the zeros of the code that support the property of hierarchical locality. As a result, we obtain a general family of hierarchical LRC codes for a new range of code parameters. We also observe that our approach enables one to represent an LRC code in quasicyclic form, and use this representation to construct tail-biting convolutional LRC codes with locality. Among other results, we extend the general approach to cyclic codes with locality to multidimensional cyclic codes, yielding new families of LRC codes with availability, and construct a family of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary cyclic hierarchical LRC codes of unbounded length.

Motivated by the recent studies on Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -additive cyclic codes, Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> [u]-cyclic codes and Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2(s)</sub> -additive codes have been introduced by Aydogdu et al.. In this paper, we study Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R-linear cyclic codes where R = Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> + uZ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , q is a prime number and u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = 0. There are two major ingredients in this paper. The first is to investigate the algebraic structure of cyclic codes and their duals over ring Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R, the spanning sets, the types and sizes of Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R-cyclic codes and their duals as well. Based on this, the second ingredient is to present an infinite family of MDSS codes and obtain some illustrative examples of q-ary cyclic codes with optimal parameters derived from the Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> R-cyclic codes.

In this chapter, we study polycyclic, negacyclic, constacyclic, quasicyclic and skew cyclic codes which are all generalizations of the important family of cyclic codes. We describe their algebraic setting and show how to use this setting to classify these families of codes.

Let Z = F_p^m[u]/(u³) be the finite commutative chain ring, where p is a prime, m is a positive integer and Fpm is the finite field with pm elements. In this paper, we determine all repeated-root constacyclic codes of arbitrary lengths over Z and their dual codes. We also determine the number of codewords in each repeated-root constacyclic code over Z. We also obtain Hamming distances, RT distances, RT weight distributions and ranks (i.e., cardinalities of minimal generating sets) of some repeated-root constacyclic codes over Z. Using these results, we also identify some isodual and maximum distance separable (MDS) constacyclic codes over Z with respect to the Hamming and RT metrics.

A (N, N-d/sub min/+1) cyclic code based on the complex-valued discrete Fourier transform is proposed. Since the code length N of the code can be any positive integer and each symbol in a code word can be a complex value (consists of two real-valued symbols), the code may be more suitable for some applications, than those codes based on Galois fields. One can design an encoder structure using the existing FFT chip. Due to the cyclic structure decoding is easy in a step-by-step manner.

On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions

Let R be the ring 𝔽2 m + u𝔽2 m, where u 2 = 0. We introduce a Gray map from R to 𝔽2 2 m and study (1 + u)-constacyclic codes over R. It is proved that the image of a (1 + u)-constacyclic code length n over R under the Gray map is a distance-invariant binary quasicyclic code of index m and length 2mn. We also prove that every code of length 2mn which is the Gray image of cyclic codes over R of length n is permutation equivalent to a binary quasi-cyclic code of index m. Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to (1 + u)-constacyclic codes over R.

Let [Formula: see text] be a Type 1 unit in [Formula: see text] where [Formula: see text] is an odd prime, [Formula: see text] is a positive integer and [Formula: see text] In this paper, we give the complete structure of all Type 1 [Formula: see text]-constacyclic codes and their duals of length [Formula: see text] over the finite commutative chain ring [Formula: see text] in terms of their generator polynomials. Using this structure, we determine the Hamming distance and the Rosenbloom–Tsfasman (RT) distance of all Type 1 [Formula: see text]-constacyclic codes. For [Formula: see text] and a unit [Formula: see text] we determine the [Formula: see text] -symbol distances of all [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] where [Formula: see text] As illustrations, we provide several [Formula: see text]-constacyclic codes with new parameters with respect to Hamming, RT and [Formula: see text]-symbol metrics. MDS codes are widely recognized for their optimal error-correction capability, and MDS [Formula: see text]-symbol codes are generalization of MDS codes. We found some MDS [Formula: see text]-symbol constacyclic codes of length [Formula: see text] over [Formula: see text] Additionally, for [Formula: see text] we provide a decoding algorithm for Type 1 constacyclic codes of length [Formula: see text] over [Formula: see text] with respect to the Hamming, RT and [Formula: see text]-symbol metrics.