Finite Fields and Their Applications最新文献

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On the construction of nonbinary LCD quadratic residue and double quadratic residue codes

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-02-11 DOI: 10.1016/j.ffa.2025.102591

Arezoo Soufi Karbaski , Taher Abualrub , Irfan Siap , Hashem Bordbar

Linear complementary dual (LCD) codes are an important class of error-correcting codes because of their applications in many areas such as their applications in cryptography and secret sharing [1], [4], [7]. In this paper, we construct a large class of nonbinary LCD codes from the class of nonbinary quadratic residue (QR) codes and nonbinary double QR codes. We have also introduced the class of extended quasi quadratic residue (QQR) codes and construct self-orthogonal codes from these codes. As an application of our study, we have presented an optimal ternary self-orthogonal code of parameters

[24, 6, 12]

. We have also constructed examples of self-orthogonal codes with parameters

{[2 (q + 1), \frac{q + 1}{2}, 2 d]}_{p}

and also examples of LCD codes with parameters

{[2 q, \frac{q - 1}{2}, 2 d]}_{p}

and

{[2 q, \frac{q + 1}{2}, 2 d]}_{p}

over

F_{p}

for different values of primes p and q.

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引用次数: 0

Homogenization of binary linear codes and their applications

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-02-06 DOI: 10.1016/j.ffa.2025.102589

Jong Yoon Hyun , Nilay Kumar Mondal , Yoonjin Lee

We introduce a new technique, called homogenization, for a systematic construction of augmented codes of binary linear codes, using the defining set approach in connection to multi-variable functions. We explicitly determine the parameters and the weight distribution of the homogenized codes when the defining set is either a simplicial complex generated by any finite number of elements, or the difference of two simplicial complexes, each of which is generated by a single maximal element. Using this homogenization technique, we produce several infinite families of optimal codes, self-orthogonal codes, minimal codes, and self-complementary codes. As applications, we obtain some best known quantum error-correcting codes, infinite families of intersecting codes (used in the construction of covering arrays), and we compute the Trellis complexity (required for decoding) for several families of codes as well.

引用次数: 0

New classes of optimal p-ary cyclic codes with minimum distance four

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-02-05 DOI: 10.1016/j.ffa.2025.102588

Zhengbang Zha , Lei Hu , Gaofei Wu

Let

C_{(1, e, s)}

denote the p-ary cyclic code with three zeros. In this paper, we present six classes of optimal p-ary cyclic codes

C_{(1, e, s)}

with parameters

[p^{n} - 1, p^{n} - 2 n - 2, 4]

by using the quadratic and quartic characters on finite fields. In addition, we propose a class of optimal quinary cyclic codes by determining the solutions of equations over

F_{5^{n}}

引用次数: 0

On matrix algebras isomorphic to finite fields and planar Dembowski-Ostrom monomials

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-02-05 DOI: 10.1016/j.ffa.2025.102590

Christof Beierle , Patrick Felke

<div><div>Let p be a prime and n a positive integer. As the first main result, we present a deterministic algorithm for deciding whether the matrix algebra <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>[</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>]</mo></math> with <math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>∈</mo><mrow><mi>GL</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math> is a finite field, performing at most <math><mi>O</mi><mo>(</mo><mi>t</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>p</mi><mo>)</mo><mo>)</mo></math> elementary operations in <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math>. In the affirmative case, the algorithm returns a defining element a so that <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>[</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>]</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>[</mo><mi>a</mi><mo>]</mo></math>.</div><div>We then study an invariant for the extended-affine equivalence of Dembowski-Ostrom (DO) polynomials. More precisely, for a DO polynomial <math><mi>g</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math>, we associate to g a set of <math><mi>n</mi><mo>×</mo><mi>n</mi></math> matrices with coefficients in <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math>, denoted <math><mrow><mi>Quot</mi></mrow><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>)</mo></math>, that stays invariant up to matrix similarity when applying extended-affine equivalence transformations to g. In the case where g is a planar DO polynomial, <math><mrow><mi>Quot</mi></mrow><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>)</mo></math> is the set of quotients <math><mi>X</mi><msup><mrow><mi>Y</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> with <math><mi>Y</mi><mo>≠</mo><mn>0</mn><mo>,</mo><mi>X</mi></math> being elements from the spread set of the corresponding commutative presemifield, and <math><mrow><mi>Quot</mi></mrow><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>)</mo></math> forms a field of order <math><msup><mrow><mi>p</

引用次数: 0

New constructions of abelian non-cyclic orbit codes based on parabolic subgroups and tensor products

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-02-05 DOI: 10.1016/j.ffa.2025.102587

Soleyman Askary, Nader Biranvand, Farrokh Shirjian

Orbit codes, as special constant dimension subspace codes, have attracted much attention due to their applications for error correction in random network coding. They arise as orbits of a subspace of

F_{q}^{n}

under the action of some subgroup of the finite general linear group

{GL}_{n} (q)

. The main contribution of this paper is to propose new methods for constructing large non-cyclic orbit codes. First, using the subgroup structure of maximal subgroups of

{GL}_{n} (q)

, we propose a new construction of an abelian non-cyclic orbit codes of size

q^{k}

with

k \leq n / 2

. The proposed code is shown to be a partial spread which in many cases is close to the known maximum-size codes. Next, considering a larger framework, we introduce the notion of tensor product operation for subspace codes and explicitly determine the parameters of such product codes. The parameters of the constructions presented in this paper improve the constructions already obtained in [6] and [7].

{"title":"New constructions of abelian non-cyclic orbit codes based on parabolic subgroups and tensor products","authors":"Soleyman Askary, Nader Biranvand, Farrokh Shirjian","doi":"10.1016/j.ffa.2025.102587","DOIUrl":"10.1016/j.ffa.2025.102587","url":null,"abstract":"<div><div>Orbit codes, as special constant dimension subspace codes, have attracted much attention due to their applications for error correction in random network coding. They arise as orbits of a subspace of <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math> under the action of some subgroup of the finite general linear group <math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math>. The main contribution of this paper is to propose new methods for constructing large non-cyclic orbit codes. First, using the subgroup structure of maximal subgroups of <math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math>, we propose a new construction of an abelian non-cyclic orbit codes of size <math><msup><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msup></math> with <math><mi>k</mi><mo>≤</mo><mi>n</mi><mo>/</mo><mn>2</mn></math>. The proposed code is shown to be a partial spread which in many cases is close to the known maximum-size codes. Next, considering a larger framework, we introduce the notion of tensor product operation for subspace codes and explicitly determine the parameters of such product codes. The parameters of the constructions presented in this paper improve the constructions already obtained in [6] and [7].</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102587"},"PeriodicalIF":1.2,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143140082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An analogue of Girstmair's formula in function fields

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-01-27 DOI: 10.1016/j.ffa.2025.102585

Daisuke Shiomi

Suppose that p is an odd prime and

g > 1

is a primitive root modulo p. Let M be a number field contained in the p-th cyclotomic field. In 1994, Girstmair found a surprising relation between the relative class number of M and the digits of

1 / p

in base g. In this paper, we consider an analogue of Girstmair's formula in function fields. Suppose that

P \in F_{q} [T]

is monic irreducible and

G \in F_{q} [T]

is a primitive root modulo P. Let L be a field extension of

F_{q} (T)

which is contained in the P-th cyclotomic function field. Our goal is to give relations between the plus and minus parts of the divisor class number of L and the digits of

1 / P

in base G.

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引用次数: 0

Fq-primitive points on varieties over finite fields

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-01-24 DOI: 10.1016/j.ffa.2025.102582

Soniya Takshak , Giorgos Kapetanakis , Rajendra Kumar Sharma

Let r be a positive divisor of

q - 1

and

f (x, y)

a rational function of degree sum d over

F_{q}

with some restrictions, where the degree sum of a rational function

f (x, y) = f_{1} (x, y) / f_{2} (x, y)

is the sum of the degrees of

f_{1} (x, y)

and

f_{2} (x, y)

. In this article, we discuss the existence of triples

(α, β, f (α, β))

over

F_{q}

, where

α, β

are primitive and

f (α, β)

is an r-primitive element of

F_{q}

. In particular, this implies the existence of

F_{q}

-primitive points on the surfaces of the form

z^{r} = f (x, y)

. As an example, we apply our results on the unit sphere over

F_{q}

引用次数: 0

The distance function on Coxeter-like graphs and self-dual codes

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-01-24 DOI: 10.1016/j.ffa.2025.102580

Marko Orel , Draženka Višnjić

<div><div>Let <math><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> be the set of all invertible <math><mi>n</mi><mo>×</mo><mi>n</mi></math> symmetric matrices over the binary field <math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. Let <math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> be the graph with the vertex set <math><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> where a pair of matrices <math><mo>{</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>}</mo></math> form an edge if and only if <math><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo>)</mo><mo>=</mo><mn>1</mn></math>. In particular, <math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub></math> is the well-known Coxeter graph. The distance function <math><mi>d</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></math> in <math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is described for all matrices <math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>. The diameter of <math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is computed. For odd <math><mi>n</mi><mo>≥</mo><mn>3</mn></math>, it is shown that each matrix <math><mi>A</mi><mo>∈</mo><msub><mrow><mi>SGL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> such that <math><mi>d</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>I</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math> and <math><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>−</mo><mi>I</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math> where I is the identity matrix induces a self-dual code in <math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math>. Conversely, each self-dual code C induces a family <math><msub><mrow><mi>F</mi></mrow><mrow><mi>C</mi></mrow></msub></math> of such matrices A. The families given by distinct self-dual codes are disjoint. The identification <math><mi>C</mi><mo>↔</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>C</mi></mrow></msub></math> provides a graph theoretical description of self-dual codes. A result of Janusz (2007) is reproved and strengthened by showing that the orthogo

引用次数: 0

MacWilliams duality for rank metric codes over finite chain rings

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-01-24 DOI: 10.1016/j.ffa.2025.102584

Iván Blanco-Chacón , Alberto F. Boix , Marcus Greferath , Erik Hieta–Aho

We extend Ravagnani's MacWilliams duality theory to the setting of rank metric codes over finite chain rings, relating the sequences of q-binomial moments of a rank metric code over this class of rings with those of its dual.

引用次数: 0

Three new classes of spreading sequence sets with low correlation and PAPR

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2025-01-22 DOI: 10.1016/j.ffa.2025.102575

Can Xiang , Chunming Tang , Wenwei Qiu

Spreading sequences have recently received a lot of attention, as some of these sequences are used to design spreading sequence sets with low correlation and low peak-to-average power ratio (PAPR for short) and which have very important applications in communication systems. It was recently reported that a small amount of work on constructing binary spreading sequence sets with low correlation and low PAPR has been done. However, till now only one work on constructing p-ary spreading sequence sets with low correlation and low PAPR for odd prime p has been done by using special functions in Liu et al. (2023) [11], and it is, in general, hard to design spreading sequence sets with low correlation and low PAPR. In this paper, we investigate this idea further by using some quadratic functions over finite fields, thereby obtain three classes of p-ary spreading sequence sets, and explicitly determine their parameters. The parameters of these p-ary spreading sequence sets are new and flexible. Furthermore, the results of this paper show that these obtained p-ary spreading sequence sets have low correlation and PAPR.

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引用次数: 0

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Finite Fields and Their Applications

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