Some New Optimal Skew Cyclic Codes With Derivation

Type 2 constacyclic codes over [formula omitted] of oddly even length

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.

In this paper, we study linear codes invariant under a cyclic endomorphism [Formula: see text] called [Formula: see text]-cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix [Formula: see text] that we call [Formula: see text]-cyclic codes, and when [Formula: see text] is the companion matrix [Formula: see text] of a given nonzero polynomial [Formula: see text] we call them [Formula: see text]-cyclic codes. The similarity relation between matrices helps us find connections between [Formula: see text]-cyclic codes and [Formula: see text]-cyclic codes, where [Formula: see text] is the minimal polynomial of [Formula: see text] The class of [Formula: see text]-cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between [Formula: see text]-cyclic codes and ideals of the polynomial ring [Formula: see text] where [Formula: see text] is the minimal polynomial of [Formula: see text] This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the [Formula: see text]-dual of an [Formula: see text]-cyclic code is an [Formula: see text]-cyclic code, where [Formula: see text] is the adjoint matrix of [Formula: see text] with respect to [Formula: see text] and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes.

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.

We compute the covering radius of each binary cyclic code of length < 64 (for both even and odd lengths) and redundancy < 28 .We also compute the covering radii of their punctured codes and shortened codes.Thus we give exact covering radii of over six thousand codes.For each of these codes (except for certain composite codes), we also determine the number of cosets of each weight less than or equal to the covering radius.These results are used to compute the minimum distances of the above cyclic codes.We use the covering radii of shortened codes and other criteria for normality to show that all but eight of the cyclic codes for which we determine the covering radius are normal.For all but seven of these normal codes, we determine the norm using some old results and some new results proved here.We observe that many cyclic codes are among the best covering codes discovered so far, and some of them lead to improvements on the previously published bounds on t[n , k], the smallest covering radius of any binary linear [n , k] code.Among some other applications of our results, we use our table of covering radii and a code augmentation argument to give four improvements on the values of dmax(n , k) .wheredmix{n, k) is the largest minimum distance of any binary [n , k] code.These results show that the covering radius is intimately connected with the other three parameters of a linear code, n, k , and d .We also give a complete classification (up to isomorphism) of cyclic self-dual codes of lengths 42, 56, and 60.The computations were carried out mainly on concurrent machines (hypercubes and Connection Machines); we give a description of our algorithm.

Let [Formula: see text] be a finite field of order [Formula: see text] and [Formula: see text] be a prime such that [Formula: see text] where [Formula: see text] are distinct odd primes with [Formula: see text] and [Formula: see text] is a positive integer with [Formula: see text]. In 2022, Rakphon et al. obtained all irreducible divisors of [Formula: see text] over [Formula: see text] where [Formula: see text] and [Formula: see text]. In this paper, we determine all primitive idempotents which can generate [Formula: see text]-constacyclic codes in [Formula: see text] where parity-check polynomials of these codes are shown in 2022. Moreover, we find the minimum Hammimg distances of the codes which are generated by primitive idempotents when [Formula: see text].

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.

The p-adic numbers were first considered by Hensel in the 19th century. He observed that the primes play an analogous role in the integers as linear polynomials do in \({\mathbb C}[X]\). The Laurent expansion of a rational function led him to consider the p-adic expansion of a rational number. In this chapter, for a fixed prime p, we will construct block codes over the rings \({\mathbb Z}/p^h{\mathbb Z}\) simultaneously, by constructing codes over the p-adic numbers and then considering the coordinates modulo ph. These codes will be linear over the ring but when mapped to codes over \({\mathbb Z}/p{\mathbb Z}\) will result in codes which are not equivalent to linear codes. We start with a brief introduction to p-adic numbers, which will cover enough background for our purposes. The classical cyclic codes, that we constructed in Chapter 5, lift to cyclic codes over the p-adic numbers. In the case of the cyclic Hamming code, this lift extends to a code over \({\mathbb Z}/4{\mathbb Z}\) which, when mapped to a binary code, gives a non-linear code with a set of parameters for which no linear code exists.

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.

Degenerated cyclic codes constitute a fascinating area of study within coding theory, offering profound insights into the realm of algebraic structures and their applications in error detection and correction. In this work, we delve into various aspects of degenerated cyclic codes, aiming to provide a comprehensive understanding of their properties and significance. We begin by elucidating the fundamental concepts underlying cyclic codes and their degeneration, establishing a mathematical framework for analysis. Subsequently, we explore the algebraic structure of degenerated cyclic codes, investigating their generator and parity-check matrices, as well as their relationships with conventional cyclic codes. Moreover, we investigate the decoding algorithms tailored for degenerated cyclic codes, evaluating their efficiency and performance under different error conditions. Furthermore, we examine the applications of degenerated cyclic codes in practical scenarios, highlighting their utility in diverse domains such as telecommunications, storage systems, and cryptography. Through theoretical analysis and numerical simulations, we demonstrate the efficacy and versatility of degenerated cyclic codes, thereby emphasizing their significance in modern information theory. Overall, this study contributes to the advancement of coding theory by shedding light on the intricacies of degenerated cyclic codes and paving the way for future research endeavors in this burgeoning field.

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.

Let \(N=p^kn\) where p is a prime, and \(k,\,n\) are positive integers satisfying \(\mathrm{gcd}(p,\,n)=1.\) We present a canonical form decomposition for every cyclic code over \(\mathbb {Z}_{p^2}\) of length N, where each subcode is concatenated by a basic irreducible cyclic code over \(\mathbb {Z}_{p^2}\) of length n as the inner code and a constacyclic code over a Galois extension ring of \(\mathbb {Z}_{p^2}\) of length \(p^k\) as the outer code. By determining their outer codes, we present a precise description for cyclic codes over \(\mathbb {Z}_{p^2}\) when \(p\ne 2,\) give precisely dual codes and investigate self-duality for cyclic codes over \(\mathbb {Z}_{p^2}.\) We end by listing cyclic self-dual codes over \(\mathbb {Z}_9\) of length 33.

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

Let p &#x2260; 3 be any prime. The structures of all &#x03BB;-constacyclic codes of length 3p^s over the finite commutative chain ring Fp^m + uFp^m (u² = 0) are established in the term of their generator polynomials. As an application, Hamming and homogeneous distance of a class of such codes and RT distances of all are given. Among such &#x03BB;-constacyclic codes, the unique maximum-distance-separable (briefly, MDS) code with respect to the RT distance is obtained. Moreover, when &#x03BB; is not a cube in Fpm, the necessary and sufficient condition for the &#x03BB;-constacyclic code of length 3p^s over Fp^m + uFp^m (u² = 0) be an MDS constacyclic code with respect to Hamming distance is provided.

Constacyclic codes over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="italic">uF</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>