Few-weight codes from trace codes over a local ring

Linear codes over F/sub p/ often admit a natural representation as trace codes of codes that are defined over an extension field F/sub p/m. In the paper, the authors obtain estimates for the weights of subcodes of such trace codes. Their main result is a far-reaching generalization of the Carlitz-Uchiyama bound for the duals of binary BCH codes. In particular, they prove sharp bounds for the generalized Hamming weights of a large class of codes, including duals of BCH codes, classical Goppa codes, Melas codes, and arbitrary cyclic codes of length n=p/sup m/-1. The main tool is the theory of algebraic functions over finite fields, in particular the Hasse-Weil bound for the number of places of degree one. >

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a proper nonnilpotent ideal in a local (Noetherian) ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a reduction of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; that is, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="subset-of-or-equal-to"> <mml:semantics> <mml:mo> ⊆ </mml:mo> <mml:annotation encoding="application/x-tex">\subseteq</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J upper I Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">JI^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="equals"> <mml:semantics> <mml:mo>=</mml:mo> <mml:annotation encoding="application/x-tex">=</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Superscript n plus 1"> <mml:semantics> <mml:msup> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">I^{n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some nonnegative integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove that there exists a finite free local unramified extension ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the ideal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I upper S"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">IS</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a minimal reduction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="subset-of-or-equal-to"> <mml:semantics> <mml:mo> ⊆ </mml:mo> <mml:annotation encoding="application/x-tex">\subseteq</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J upper S"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">JS</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the property that the number of elements in a minimal generating set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-te

Mixed alphabet codes are the generalizations of the classical linear codes over finite fields and rings. In this letter we introduce a valid method of constructing \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> R-additive codes, where p is a prime number, and R = \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> +u \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> with u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> =0. These codes can be viewed as a combination of irreducible cyclic codes over \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> and trace codes over R. In particular, by using the Gray map, part of the obtained image codes meets the Greismer bound with equality, and can be applied to construct new secret sharing scheme.

Algebraic function fields (or equivalently, algebraic curves) provide a useful tool for coding theory (for instance, algebraic-geometric codes and trace codes), but also for other branches of information theory. In these applications, the number of rational places of a function field plays a crucial role. One is particularly interested in function fields having a large number of rational places. After a short introduction into the mathematical theory of algebraic functions, the paper gives a survey of old and new results on the number of rational places of function fields.

No AccessTwisted Hessian curves over the ring 𝔽q[∈], ∈2 = 0Abdelâli Grini, Abdelhakim Chillali, Lhoussain El Fadil and Hakima MouanisAbdelâli GriniDepartment of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco, Abdelhakim ChillaliDepartment of Mathematics, Physical and Computer Sciences, FP, LSI, Taza, S.M. Ben Abdellah University, Fez, Morocco, Lhoussain El FadilDepartment of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco and Hakima MouanisDepartment of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, MoroccoPublished Online:November 28, 2022pp 181-189https://doi.org/10.1504/IJCAET.2023.127795PDF ToolsAdd to FavouritesDownload CitationsTrack Citations Share this article on social mediaShareShare onFacebookTwitterLinkedInReddit AboutAbstractThe goal of this work is to study some arithmetic proprieties of the twisted Hessian curves defined by a equation of type: aX3 + Y3 + Z3 = dXYZ on the local ring R2 = 𝔽q[X]/(X2), where p ≥ 5 is a prime number, q = pd and d ∈ ℕ*, such that -3 is not a square in 𝔽q. This paper consists of an introduction, section, and a conclusion. In the introduction, we review some fundamental arithmetic proprieties of finite local rings R2, which will be used in the remainder of this article. The section is devoted to the study the above mentioned twisted Hessian curves on these finite local rings for restriction to some specific characteristic p ≥ 5. Using these studies, we give essential properties and we define the group H2a,d, these properties, the classification of these elements and a bijection between the sets H2a,d and Ha0,d0 × 𝔽q, where Ha0,d0 is the twisted Hessian curve over the finite field 𝔽q.Keywordstwisted Hessian curves, finite over ring, cryptography, elliptic curves, local ring Previous article Next article FiguresReferencesRelatedDetails Volume 18Issue 1-32023 ISSN: 1757-2657eISSN: 1757-2665 HistoryPublished onlineNovember 28, 2022 Copyright © 2023 Inderscience Enterprises Ltd.Keywordstwisted Hessian curvesfinite over ringcryptographyelliptic curveslocal ringAuthors and AffiliationsAbdelâli Grini1 Abdelhakim Chillali2 Lhoussain El Fadil3 Hakima Mouanis4 1. Department of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco2. Department of Mathematics, Physical and Computer Sciences, FP, LSI, Taza, S.M. Ben Abdellah University, Fez, Morocco3. Department of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, Morocco4. Department of Mathematics, Faculty of Science Dhar El Mahraz-Fez, S.M. Ben Abdellah University, Fez, MoroccoPDF download

Let \({\mathcal{O}}\) be the local ring at a singular point of a geometrically integral algebraic curve defined over a finite field, and let m be the number of branches centered at the curve singularity. By encoding cardinalities of certain finite sets of ideals, we associate to each pair of ideal classes of \({\mathcal{O}}\) a power series in m variables with integer coefficients, which can be represented by an integral within the framework of harmonic analysis. We prove that partial local zeta functions can be expressed in terms of these multi-variable power series. The main objective of this paper is to investigate the properties of these series, and to provide in this way a deeper insight into the nature of local zeta functions.

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.

Tim Blackmore and Graham H. Norton introduced the notion of matrix-product codes over finite fields. The present paper provides a generalization to finite chain rings. For codes a distance function is defined using a homogeneous weight function in the ring. It is proved that the minimum distance of a matrix-product codes is determined by the minimum distances of the separate codes. At the end of the paper we focus on Galois rings and define a special family of matrix-product codes.

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ding et al. (IEEE Trans Inform Theory 57(12), 8000–8006, 2011); Ma et al. (IEEE Trans Inform Theory 57(1):397–402, 2011); Wang et al. (Trans Inf Theory 58(12):7253–7259, 2012); and Xiong (Finite Fields Appl 18(5):933–945, 2012). In this paper we use the method developed in Xiong (Finite Fields Appl 18(5):933–945, 2012) to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which turns out to be associated with counting the number of points on some elliptic curves over finite fields. We also treat the special case that the characteristic of the finite field is 2.

On the structure of repeated-root polycyclic codes over local rings

Generalized Loewy length of Cohen-Macaulay local and graded rings

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let ? be a Galois ring of characteristic 4,X a cyclotomic scheme over ? with classD and ? the Terwilliger algebra ofX, then we show that most of the irreducible ?-modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible ?-modules using Jacobi sums.

The generalized chain geometry over the local ring $$K(\varepsilon ; \sigma)$$ of twisted dual numbers, where K is a finite field, is interpreted as a divisible design obtained from an imprimitive group action. Its combinatorial properties as well as a geometric model in 4-space are investigated.

In this paper, we present several doubly infinite families of linear projective codes with two-, three- and five distinct non-zero Hamming weights together with the frequency distributions of their weights.

An (n, k, l) maximum distance separable (MDS) array code of length n, dimension k = n - r, and subpacketization l is formed of l × n matrices over a finite field F, with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword. Repair of a failed node (recovery of one erased column) can be performed by accessing a set of d ≤ n - 1 surviving (helper) nodes. The code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity. For optimal-access MDS codes with d = n-1, the sub-packetization l satisfies the bound l ≥ r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k-1)/r</sup> . In our previous work (IEEE Trans. Inf. Theory, vol. 63, no. 4, 2017), for any n and r, we presented an explicit construction of optimal-access MDS codes with subpacketization l = r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⌈n-1⌉</sup> . In this paper, we take up the question of reducing the sub-packetization value l to make it to approach the lower bound. We construct an explicit family of optimal-access codes with l = r1.n/rl, which differs from the optimal value by at most a factor of r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . These codes can be constructed over any finite field F as long as |F| ≥ r⌈n/r⌉, and afford low-complexity encoding and decoding procedures. We also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints (LRC codes), which we call group repair with optimal access. In this variation, we assume that the set of n = sm nodes is partitioned into m repair groups of size s, and require that the amount of accessed data for repair is the smallest possible whenever the d = s + k - 1 helper nodes include all the other s-1 nodes from the same group as the failed node. For this problem, we construct a family of codes with the group optimal access property. These codes can be constructed over any field F of size |F| ≥ n, and also afford low-complexity encoding and decoding procedures.