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On dual-containing, almost dual-containing matrix-product codes and related quantum codes Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-03-13 Meng Cao
Matrix-product (MP) codes are a type of long codes formed by combining several commensurate constituent codes with a defining matrix. In this paper, we study the MP code when the defining matrix satisfies the condition that is -monomial. We give an explicit formula for calculating the dimension of the hull of a MP code. We present the necessary and sufficient conditions for a MP code to be dual-containing
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The extended codes of some linear codes Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-03-12 Zhonghua Sun, Cunsheng Ding, Tingfang Chen
The classical way of extending an linear code is to add an overall parity-check coordinate to each codeword of the linear code . This extended code, denoted by and called the standardly extended code of , is a linear code with parameters , where or . This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes
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Non-Abelian extensions of degree p3 and p4 in characteristic p > 2 Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-03-12 Grant Moles
This paper describes in terms of Artin-Schreier equations field extensions whose Galois group is isomorphic to any of the four non-cyclic groups of order or the ten non-Abelian groups of order , an odd prime, over a field of characteristic .
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Diagonal hypersurfaces and elliptic curves over finite fields and hypergeometric functions Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-03-05 Sulakashna, Rupam Barman
Let denote the family of diagonal hypersurface over a finite field given by where , , and . Let denote the number of points on in . It is easy to see that is equal to the number of distinct zeros of the polynomial in . In this article, we prove that is also equal to the number of distinct zeros of the polynomial in . We express the number of distinct zeros of the polynomial in terms of a -adic hypergeometric
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A structure theorem for the restricted sum of four squares Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-03-01 Wei Wang, Weijia Wang, Hao Zhang
Let be an odd prime. We show that each solution of the system of congruence equations and corresponds to precisely four solutions of the system of Diophantine equations and that are pairwise orthogonal over , partially answering a conjecture proposed in Wang et al. . The result was obtained by counting the number of solutions of both equations using Gaussian sum and modular forms, and the classical
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A class of double-twisted generalized Reed-Solomon codes Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-03-01 Canze Zhu, Qunying Liao
In this paper, let be a prime power, we focus on a class of double-twisted generalized Reed-Solomon code over . We give a sufficient and necessary condition for to be MDS or AMDS, and prove that is non-GRS by calculating the Schur square of its dual code. Furthermore, we present a sufficient and necessary condition for to be self-dual, and then construct several classes of self-dual NMDS or non-GRS
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Transverse linear subspaces to hypersurfaces over finite fields Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-28 Shamil Asgarli, Lian Duan, Kuan-Wen Lai
Ballico proved that a smooth projective variety of degree and dimension over a finite field of elements admits a smooth hyperplane section if . In this paper, we refine this criterion for higher codimensional linear sections on smooth hypersurfaces and for hyperplane sections on Frobenius classical hypersurfaces. We also prove a similar result for the existence of reduced hyperplane sections on reduced
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Some new classes of additive MDS and almost MDS codes over finite fields Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-27 Monika Yadav, Anuradha Sharma
In this paper, we introduce and study two new classes of additive codes over finite fields, additive generalized Reed-Solomon (additive GRS) codes and additive generalized twisted Reed-Solomon (additive GTRS) codes, which are extensions of linear generalized Reed-Solomon (GRS) codes and twisted Reed-Solomon (GTRS) codes, respectively. Unlike linear GRS codes, additive GRS codes are not maximum distance
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Constructions and decoding of GC-balanced codes for edit errors Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-27 Kenan Wu, Shu Liu
DNA-based storage has been a promising technique of data storage, due to its high density and long duration. During synthesizing and sequencing of DNA storage, edit errors including insertions, deletions and substitutions are introduced inevitably. An effective way to reduce the error probability is to limit the content of G and C in DNA sequences to around 50%, which is called GC-balanced. To deal
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On m-ovoids of Q+(7,q) with q odd Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-22 Sam Adriaensen, Jan De Beule, Giovanni Giuseppe Grimaldi, Jonathan Mannaert
In this paper, we provide a construction of -ovoids of the hyperbolic quadric , an odd prime power, by glueing -ovoids of the elliptic quadric . This is possible by controlling some intersection properties of (putative) -ovoids of elliptic quadrics. It eventually yields -ovoids of not coming from a 1-system. Secondly, for certain values of , we construct line spreads of that have as many secants to
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On subgroup perfect codes in Cayley sum graphs Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-22 J, u, n, y, a, n, g, , Z, h, a, n, g
A perfect code in a graph Γ is an independent set of vertices of Γ such that every vertex outside is adjacent to a unique vertex in , and a total perfect code in Γ is a set of vertices of Γ such that every vertex of Γ is adjacent to a unique vertex in . Let be a finite group and a normal subset of . The Cayley sum graph of with the connection set is the graph with vertex set and two vertices and being
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AG codes on flag bundles over a curve Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-22 T, o, h, r, u, , N, a, k, a, s, h, i, m, a
In the present paper, we construct codes from the flag bundle associated to a vector bundle over a curve. Our code may be considered as a relative version of the codes on the flag variety studied by F. Rodier. We investigate the dimension and the minimum distance of such relative Rodier codes using intersection theory. For this purpose, we exploit the invariants of vector bundles which control the
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Saturating linear sets of minimal rank Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-21 Daniele Bartoli, Martino Borello, Giuseppe Marino
Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this paper, we consider the recently introduced linear version of such sets, which is, in turn, related to the covering problem in the rank metric. The main questions
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Construction of εd-ASIC-POVMs via 2-to-1 PN functions and the Li bound Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-20 Meng Cao, Xiantao Deng
Symmetric informationally complete positive operator-valued measures (SIC-POVMs) in finite dimension are a particularly attractive case of informationally complete POVMs (IC-POVMs), which consist of subnormalized projectors with equal pairwise fidelity. However, it is difficult to construct SIC-POVMs, and it is not even clear whether there exists an infinite family of SIC-POVMs. To realize some possible
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Weight distribution of double cyclic codes over [formula omitted] Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-20 Xiangrui Meng, Jian Gao, Fang-Wei Fu
In this paper, we determine the Hamming weight distribution of several classes of double cyclic codes over the finite non-chain ring with , which is isomorphic to . Some two-weight and three-weight linear codes over the finite field are constructed by double cyclic codes over this ring.
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Twisted group algebras of Abelian groups Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-15 André Duarte, Raul Antonio Ferraz, César Polcino Milies
We study twistings for finite Abelian groups over fields and then show how to extend the notion of idempotent determined by a subgroup, so useful in the case of group algebras, to the case of twisted group algebras, at least when the subgroup is cyclic. In doing so, we obtain a method to compute in a direct way the primitive central idempotents of these algebras over a finite field and fully describe
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Enveloping sieve related to the Hardy-Littlewood irreducible tuple conjecture in a function field Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-09 G, u, o, q, u, a, n, , L, i
Let be the polynomial ring over the finite field of elements. For , let with . Let denote the product of linear functions . Let be the set of all monic polynomials in such that is the product of monic irreducible polynomials in . The Hardy-Littlewood -tuple conjecture provides an asymptotic formula for the cardinality of the set when tends to ∞. In this paper an -version of the enveloping sieve theory
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Codes over a ring of order 32 with two Gray maps Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-09 S.T. Dougherty, J. Gildea, A. Korban, A.M. Roberts
We describe a ring of order 32 and prove that it is a local Frobenius ring. We study codes over this ring and we give two distinct non-equivalent linear orthogonality-preserving Gray maps to the binary space. Self-dual codes are studied over this ring as well as the binary self-dual codes that are the Gray images of those codes. Specifically, we show that the image of a self-dual code over this ring
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Characterizations of a class of planar functions over finite fields Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-08 Ruikai Chen, Sihem Mesnager
Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form on an extension of the finite field . Specifically, we investigate those functions on and construct several typical kinds of planar functions. We also completely characterize them on . When the degree of extension is higher
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New constructions of optimal binary LCD codes Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-08 Guodong Wang, Shengwei Liu, Hongwei Liu
Linear complementary dual (LCD) codes provide an optimum linear coding solution for the two-user binary adder channel. LCD codes also can be used against side-channel attacks and fault non-invasive attacks. In this paper, we obtain a lower bound on the distance of binary LCD codes through expanded codes. We give necessary and sufficient conditions to extend binary LCD codes to binary and LCD codes
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Exceptionality of (X+1)n + (X−1)n Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-05 Zhiguo Ding, Michael E. Zieve
We determine all prime powers and all positive integers for which and permutes . We also exhibit instances of these polynomials which are closely related to Dickson polynomials, and we give a similar connection between Dickson polynomials and the polynomials .
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Zero testing and equation solving for sparse polynomials on rectangular domains Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-02 Erhard Aichinger, Simon Grünbacher, Paul Hametner
We consider sparse polynomials in variables over a finite field, and ask whether they vanish on a set , where is a set of nonzero elements of the field. We see that if for a polynomial , there is ▪ with ▪, then there is such a ▪ in every ball (with respect to the Hamming distance) inside , where the radius of the ball is bounded by a multiple of the logarithm of the number of monomials that appear
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Hausdorff dimension for sets of continued fractions of formal Laurent series Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-01 Mumtaz Hussain, Nikita Shulga
We prove the Hausdorff dimension of various limsup sets over the field of formal power series. Typically, the upper bound is easier to establish by considering the natural covering of the underlying set. To establish the lower bound, we identify a suitable set that serves as a subset of several limsup sets by selecting appropriate values for the involved parameters. To be precise, given a fixed integer
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On LCD codes from skew symmetric Toeplitz matrices Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-01 K, a, i, m, i, n, , C, h, e, n, g
A linear code with complementary dual (or an LCD code) is defined to be a linear code which intersects its dual code trivially. Let be an identity matrix and be a Toeplitz matrix of the same order over a finite field. A Double Toeplitz code (or a DT code) is a linear code generated by a generator matrix of the form . In 2021, Shi et al. obtained necessary and sufficient conditions for a Double Toeplitz
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Several classes of permutation polynomials based on the AGW criterion over the finite field [formula omitted] Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-01 Guanghui Li, Xiwang Cao
Let denote the finite field with elements. Permutation polynomials and complete permutation polynomials over finite fields have been widely investigated in recent years due to their applications in cryptography, coding theory and combinatorial design. In this paper, several classes of (complete) permutation polynomials with the form and are proposed based on the AGW criterion and some techniques in
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Superzeta functions on function fields Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-02-01 Kajtaz H. Bllaca, Jawher Khmiri, Kamel Mazhouda, Bouchaïb Sodaïgui
We study the superzeta functions on function fields as constructed by Voros (see ) in the case of the classical Riemann zeta function. Furthermore, we study special values of those functions, relate them to the Li coefficients, deduce some interesting summation formulas, and prove some results about the regularized product of the zeros of zeta functions on function fields.
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A new direction on constructing irreducible polynomials over finite fields Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-25 K, a, i, m, i, n, , C, h, e, n, g
Let be a power of a prime and be the finite field of order . Let be any polynomial in and . For any positive integer , denote to be the -th iterate of Φ and to be the denominator of . We call inversely stable over if are distinct and irreducible over for all . In this paper, we aim to find a class of inversely stable polynomials over . Actually, let , it is proved that is inversely stable over if and
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A note on t-designs in isodual codes Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-24 Madoka Awada, Tsuyoshi Miezaki, Akihiro Munemasa, Hiroyuki Nakasora
In the present paper, we construct 3-designs using extended binary quadratic residue codes and their dual codes.
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New constructions of optimal (r,δ)-LRCs via good polynomials Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-24 Yuan Gao, Siman Yang
Locally repairable codes (LRCs) are a class of erasure codes that are widely used in distributed storage systems, which allow for efficient recovery of data in the case of node failures or data loss. In 2014, Tamo and Barg introduced Reed-Solomon-like (RS-like) Singleton-optimal -LRCs based on polynomial evaluation. These constructions rely on the existence of so-called good polynomial that is constant
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Constructing permutation polynomials over [formula omitted] from bijections of PG(2,q) Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-23 Longjiang Qu, Kangquan Li
Over the past several years, there are numerous papers about permutation polynomials of the form over . A bijection between the multiplicative subgroup of and the projective line plays a very important role in the research. In this paper, we mainly construct permutation polynomials of the form over from bijections of the projective plane . A bijection from the multiplicative subgroup of to is studied
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Permutation rational functions over quadratic extensions of finite fields Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-23 Ruikai Chen, Sihem Mesnager
Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over , whose numerators are so-called -quadratic polynomials. To this end, we will first determine the exact number of zeros of a special -quadratic polynomial in , by calculating character sums related to quadratic forms of . Then given
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Matrices in [formula omitted] with quadratic minimal polynomial Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-16 J, a, c, o, b, u, s, , V, i, s, s, e, r, , v, a, n, , Z, y, l
By a result of Latimer and MacDuffee, there are a finite number of equivalence classes of matrices over with minimum polynomial , where is an degree polynomial, irreducible over . In this paper, we develop an algorithm for finding a canonical representative of each matrix class, for .
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F-valued trace of a finite-dimensional commutative F-algebra Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-15 A.K. Bhagat, R. Sarma
A non-zero F-valued F-linear map on a finite-dimensional commutative F-algebra is called an F-valued trace if its kernel contains no non-zero ideals. However, given an F-algebra, such a map may not always exist. We find an infinite class of finite-dimensional commutative F-algebras which admit an F-valued trace. In fact, in these cases, we explicitly construct a trace map. An F-valued trace on a finite-dimensional
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Locally recoverable codes from towers of function fields Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-08 M. Chara, F. Galluccio, E. Martínez-Moro
In this work we construct sequences of locally recoverable AG codes arising from a tower of function fields and give bound for the parameters of the obtained codes. In a particular case of a tower over Fq2 for any odd q, defined by Garcia and Stichtenoth in [3], we show that the bound is sharp for the first code in the sequence, and we include a detailed analysis for the following codes in the sequence
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Bilinear forms with Kloosterman and Gauss sums in function fields Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-08 Christian Bagshaw
In recent years, there has been a lot of progress in obtaining non-trivial bounds for bilinear forms of Kloosterman sums in Z/mZ for arbitrary integers m. These results have been motivated by a wide variety of applications, such as improved asymptotic formulas for moments of L-functions. However, there has been very little work done in this area in the setting of rational function fields over finite
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On the eigenvalues of the graphs D(5,q) Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-03 Himanshu Gupta, Vladislav Taranchuk
Let q=pe, where p is a prime and e is a positive integer. The family of graphs D(k,q), defined for any positive integer k and prime power q, were introduced by Lazebnik and Ustimenko in 1995. To this day, the connected components of the graphs D(k,q), provide the best known general lower bound for the size of a graph of given order and given girth. Furthermore, Ustimenko conjectured that the second
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Hermitian self-dual 2-quasi-abelian codes Finite Fields Their Appl. (IF 1.0) Pub Date : 2024-01-04 Guanghui Zhang, Liren Lin, Chunyan Qin, Ruibo Li
In this paper, we construct a class of Hermitian self-dual 2-quasi-abelian codes over a finite field. Based on counting the number of such codes and estimating the number of the codes in this class whose relative minimum weights are small, we prove that the class of Hermitian self-dual 2-quasi-abelian codes over any finite field is asymptotically good. The existence of such codes is unconditional,
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Subfield subcodes of projective Reed-Muller codes Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-12-27 Philippe Gimenez, Diego Ruano, Rodrigo San-José
Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective space, we generalize the necessary tools to deal with this case as well: we obtain a universal Gröbner basis for the vanishing ideal of the set of standard representatives
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Two classes of permutation trinomials over Fq3 in characteristic two Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-12-20 Lijing Zheng, Haibin Kan, Tongliang Zhang, Jie Peng, Yanjun Li
Let q=2m and Fq3 be the finite field with q3 elements. In this paper, based on the multivariate method, resultant elimination, and transforming into dealing with some equations over finite fields, we propose two classes of permutation trinomials of Fq3. We illustrate that these two classes of permutation trinomials are QM-inequivalent to all known permutation polynomials over Fq3. Some well-known results
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Fourth power mean of the general s-dimensional Kloosterman sum mod p Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-12-15 Nilanjan Bag, Anup Haldar
In this article, we prove an asymptotic formula for the fourth power mean of a general s-dimensional hyper-Kloosterman sum. We find the number of solutions of certain congruence equations mod p which play an integral part to prove our main result. We use estimates for character sums and analytic methods to prove our theorem.
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Codes and pseudo-geometric designs from the ternary m-sequences with Welch-type decimation d = 2 ⋅ 3(n−1)/2 + 1 Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-12-06 Can Xiang, Chunming Tang, Haode Yan, Min Guo
Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the methods of finite geometries and combinatorics. However, none of pseudo-geometric designs with the parameters S(2,q+1,(qn−1)/(q−1)) is constructed by the approach of coding
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Counting roots of fully triangular polynomials over finite fields Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-12-05 José Gustavo Coelho, Fabio Enrique Brochero Martínez
Let Fq be a finite field with q elements, f∈Fq[x1,…,xn] a polynomial in n variables and let us denote by N(f) the number of roots of f in Fqn. In this paper we consider the family of fully triangular polynomials, i.e., polynomials of the formf(x1,…,xn)=a1x1d1,1+a2x1d1,2x2d2,2+…+anx1d1,n⋯xndn,n−b, where di,j>0 for all 1≤i≤j≤n. For these polynomials, we obtain explicit formulas for N(f) when the augmented
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Grassmannians of codes Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-12-01 Ilaria Cardinali, Luca Giuzzi
Consider the point line-geometry Pt(n,k) having as points all the [n,k]-linear codes having minimum dual distance at least t+1 and where two points X and Y are collinear whenever X∩Y is a [n,k−1]-linear code having minimum dual distance at least t+1. We are interested in the collinearity graph Λt(n,k) of Pt(n,k). The graph Λt(n,k) is a subgraph of the Grassmann graph and also a subgraph of the graph
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The isomorphism problem in the context of PI-theory for two-dimensional Jordan algebras Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-12-01 Diogo Diniz, Dimas José Gonçalves, Viviane Ribeiro Tomaz da Silva, Manuela da Silva Souza
Let F be a field of characteristic different from 2. Small-dimensional Jordan algebras over F have been extensively studied and classified. In the present paper we show that any two-dimensional Jordan algebras over a finite field are isomorphic if and only if they satisfy the same polynomial identities (the opposite happens in the case F is infinite, even if algebraically closed). We determine a finite
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Remarks on the inverse Galois problem over function fields Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-11-24 Shiang Tang
In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of ℓ-adic Galois representations valued in a semisimple group G using Galois theoretic and automorphic methods, and then proving that the Galois images are maximal for a set of primes of positive density using a classical result
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Note on Budaghyan and Carlet's almost perfect nonlinear functions Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-11-24 Huan Sun, Qin Yue, Xue Jia
Almost perfect nonlinear (APN) functions have good properties and are widely applied in sequence design and coding theory. Budaghyan and Carlet (2008) [5] constructed a family of APN hexanomials F3 over F22m with a certain technical condition. In this article, we give the number of APN hexanomials F3 and support a determination theorem for APN hexanomials F3 if i=1. Moreover, we construct a family
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Near-MDS codes from maximal arcs in PG(2,q) Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-11-22 Li Xu, Cuiling Fan, Dongchun Han
The singleton defect of an [n,k,d] linear code C is defined as s(C)=n−k+1−d. Codes with S(C)=0 are called maximum distance separable (MDS) codes, and codes with S(C)=S(C⊥)=1 are called near maximum distance separable (NMDS) codes. Both MDS codes and NMDS codes have good representations in finite projective geometry. MDS codes over Fq with length n and n-arcs in PG(k−1,q) are equivalent objects. When
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On CCZ-equivalence between the Bracken-Tan-Tan function and power functions Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-11-20 Chenmiao Shi, Jie Peng, Haibin Kan, Lijing Zheng
Permutations with differential and boomerang uniformity 4 over F22k offer good resistance to block ciphers against differential and boomerang attacks. There are five primarily constructed infinite classes of differentially 4-uniform permutations with the best known nonlinearity over F22k, namely the Gold functions, the Kasami functions, the Inverse functions, the Bracken-Leander functions and the Bracken-Tan-Tan
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Some results on complete permutation polynomials and mutually orthogonal Latin squares Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-11-07 Chandan Kumar Vishwakarma, Rajesh P. Singh
In this paper, we investigate some classes of complete permutation polynomials (CPPs) with the form (L1(x))t+L2(x) for some specific linearized polynomials L1(x) and L2(x) over finite fields. Some constructions of PPs and CPPs over finite fields using the AGW criterion are also proposed. We also obtain some constructions of sets of Mutually orthogonal Latin squares (MOLS) using permutation polynomials
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Weight enumerators of all cubic-primitive irreducible cyclic codes of odd prime power length Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-11-06 Monika Bishnoi, Pankaj Kumar
Let p and q be odd primes and q be a cubic primitive modulo pu for some positive integer u. In this paper, we prove that the solutions of some Diophantine equations provide the weight enumerators of some cubic primitive irreducible cyclic codes of prime length. Bounds on the minimum distances of these codes are also given.
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Character sums over elements of extensions of finite fields with restricted coordinates Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-11-01 Siddharth Iyer, Igor E. Shparlinski
We obtain nontrivial bounds for character sums with multiplicative and additive characters over finite fields over elements with restricted coordinate expansion. In particular, we obtain a nontrivial estimate for such a sum over a finite field analogue of the Cantor set.
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On the distribution of the entries of a fixed-rank random matrix over a finite field Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-10-28 Carlo Sanna
Let r>0 be an integer, let Fq be a finite field of q elements, and let A be a nonempty proper subset of Fq. Moreover, let M be a random m×n rank-r matrix over Fq taken with uniform distribution. We prove, in a precise sense, that, as m,n→+∞ and r,q,A are fixed, the number of entries of M that belong to A approaches a normal distribution.
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More on the DLW conjectures Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-10-28 Daniele Bartoli, Matteo Bonini
We prove two conjectures involving permutation polynomials in a paper of Dmytrenko, Lazebnik, Williford, in a low degree regime, using the theory of algebraic curves over finite fields. More precisely, we prove that Conjecture A holds whenever q≥max{(2k−1)2+1,1.823(4k2−14k+12)}, whereas Conjecture B holds if q≥2.233(9k2−21k+12). Although one of these conjectures was already proved by Hou without any
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Polarization-adjusted convolutional (PAC) codes as a concatenation of inner cyclic and outer polar- and Reed-Muller-like codes Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-10-23 Mohsen Moradi
Polarization-adjusted convolutional (PAC) codes are a new family of linear block codes that can perform close to the theoretical bounds in the short block-length regime. These codes combine polar coding and convolutional coding. In this study, we show that PAC codes are equivalent to a new class of codes consisting of inner cyclic codes and outer polar- and Reed-Muller-like codes. We leverage the properties
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Some new results on permutation polynomials of the form b(xq+ax+δ)s−ax over Fq2 Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-10-24 Danyao Wu, Pingzhi Yuan
This paper investigates the permutation behavior of polynomials in the form b(xpm+ax+δ)s−ax over Fp2m by employing the Akbary–Ghioca–Wang criterion, where b∈Fpm⁎, a,δ∈Fp2m with aq+1=1. As a result, we present several classes of permutation polynomials of the form b(x2m+ax+δ)s+ax over F22m and three classes of permutation polynomials of the form b(xpm−x+δ)s+x over Fp2m with infinitely many odd primes
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Multi-twisted additive self-orthogonal and ACD codes are asymptotically good Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-10-23 Sandeep Sharma, Anuradha Sharma
Multi-twisted (MT) additive codes over finite fields constitute an important class of additive codes and are generalizations of cyclic and constacyclic additive codes. In this paper, we employ probabilistic methods and results from groups and geometry to study the asymptotic behavior of the rates and relative Hamming distances of two special subclasses of MT additive codes over finite fields, viz.
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On the distribution of pseudorandom vectors generated by elliptic curves Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-10-18 Xiaoyu Wang
In the present paper, the distribution of pseudorandom vectors, derived from a specific set of rational points on elliptic curves over finite fields, is studied by estimating its discrepancy based on the system of Walsh functions and exponential sums over elliptic curves. It turns out these pseudorandom vectors have good behaviors, such as uniform distribution and strong pseudorandomness. Moreover
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Partial permutation decoding and PD-sets for Zps-linear generalized Hadamard codes Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-10-20 Adrián Torres-Martín, Mercè Villanueva
It is known that Zps-linear codes, which are the Gray map image of Zps-additive codes (linear codes over Zps), are systematic and a systematic encoding has been found. This makes Zps-linear codes suitable to apply the permutation decoding method. This technique is also based on the existence of r-PD-sets, which are subsets of the permutation automorphism group of the code. In this paper, we study the
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A tight upper bound on the number of non-zero weights of a constacyclic code Finite Fields Their Appl. (IF 1.0) Pub Date : 2023-10-20 Hanglong Zhang, Xiwang Cao
For a simple-root λ-constacyclic code C over Fq, let 〈ρ〉 and 〈ρ,M〉 be the subgroups of the automorphism group of C generated by the cyclic shift ρ, and by the cyclic shift ρ and the scalar multiplication group M, respectively. Let NG(C⁎) be the number of orbits of a subgroup G of the automorphism group of C acting on C⁎=C﹨{0}. In this paper, we establish explicit formulas for N〈ρ〉(C⁎) and N〈ρ,M〉(C⁎)