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Partial spreads are important in finite geometry and can be used to construct linear codes. From the results in (Des. Codes Cryptogr. 90, 1–15, 2022) by Xia Li, Qin Yue and Deng Tang, we know that if the number of the elements in a partial spread is "big enough", then the corresponding linear code is minimal. This paper used the sufficient condition in (IEEE Trans. Inf. Theory 44(5), 2010–2017, 1998) to prove the minimality of such linear codes. In the present paper, we use the geometric approach to study the minimality of linear codes constructed from partial spreads in all cases. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) codes, which are a class of linear codes introduced by Massey, have been extensively studied in the literature recently. It has been shown that LCD codes play a role in measures to counter passive and active side-channel analyses on embedded cryptosystems. In this paper, tables are presented of good quaternary Hermitian LCD codes and they are used in the construction of puncturing, shortening and combination codes. The results of this, including three tables of the best-known quaternary Hermitian LCD codes of any length n ≤ 25 with corresponding dimension k, are presented. In addition, many of these quaternary Hermitian LCD codes given in this paper are optimal and saturate the lower or upper bound of Grassl's code table, and some of them are nearly optimal. [ABSTRACT FROM AUTHOR]

In this paper we will analyze one linear code from the theoretical point of view. Namely, the code definition is based on linear quasigroups. In the previous work we classified the quasigroups of order 4 according to their probability of undetected errors. Now, in this paper we will conclude whether the linear quasigroups that are in the same class in this classification obtain equal number of surely detected incorrectly transmitted bits. Also, we will classify the linear quasigroups of order 4 according to the number of errors that the code surely detects when they are used for coding. At the end we will make conclusion which quasigroups of order 4 are overall best for coding having in mind both important parameters for every code for error detection: the number of errors that the code surely detects and the probability of undetected errors. [ABSTRACT FROM AUTHOR]

Reversible codes have a range of wide applications in cryptography, data storage, and communication systems. In this paper, we investigated reversible codes under the Rosenbloom-Tsfasman metric (RT-metric). First, some properties of reversible codes in the RT-metric were described. An essential condition for a reversible code to be a maximum distance separable code (MDS code, in short) in the RT-metric was established. A necessary condition for a binary self-dual code to be reversible was proven and the same was generalized for q-ary self-dual reversible codes. Several constructions for reversible RT-metric codes were provided in terms of their generator matrices. [ABSTRACT FROM AUTHOR]

In this paper, three discrete transforms related to vector spaces over finite fields are studied. For our purposes, and according to the properties of the finite fields, the most suitable transforms are as follows: for binary fields, this is the Walsh–Hadamard transform; for odd prime fields, the Vilenkin–Chrestenson transform; and for composite fields, the trace transform. A factorization of the transform matrices using Kronecker power is given so that the considered discrete transforms are reduced to the fast discrete transforms. Examples and applications are also presented of the considered transforms in coding theory for calculating the weight distribution of a linear code. [ABSTRACT FROM AUTHOR]

Let R be the Galois ring of characteristic 4 and cardinality 4 m , where m is a natural number. Let C be a linear code of length n over R and Φ be the Homogeneous Gray map on R n . In this paper, we show that Φ (C) is linear if and only if for every X , Y ∈ C , 2 (X ⊙ Y) ∈ C . Using the generator polynomial of a cyclic code of odd length over R, a necessary and sufficient condition is given which its Gray image is linear. [ABSTRACT FROM AUTHOR]

The hull of a linear code plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. Regarding the quantum error correction, linear codes with determined hull are used to construct quantum codes. In this paper, we focus on the hull of Simplex codes and punctured Simplex codes. We firstly study the properties of the matrix produced by the column vectors of a projective space and determine the Euclidean and Hermitian hull of punctured Simplex codes completely. Secondly, we investigate the Euclidean and Hermitian hull of several classes of linear codes from Simplex codes. [ABSTRACT FROM AUTHOR]

Side-channel attacks aim at extracting secret keys from cryptographic devices. Randomly masking the implementation is a provable way to protect the secrets against this threat. Recently, various masking schemes have converged to the "code-based masking" philosophy. In code-based masking, different codes allow for different levels of side-channel security. In practice, for a given leakage function, it is important to select the code which enables the best resistance, i.e., which forces the attacker to capture and analyze the largest number of side-channel traces. This paper is a first attempt to address the constructive selection of the optimal codes in the context of side-channel countermeasures, in particular for code-based masking when the device leaks information in the Hamming weight leakage model. We show that the problem is related to the weight enumeration of the extended dual of the masking code. We first present mathematical tools to study those weight enumeration polynomials, and then provide an efficient method to search for good codes, based on a lexicographic sorting of the weight enumeration polynomial from the lowest to highest degrees. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new class of error-detecting codes based on quasigroups using automorphisms of the finite field 픽pn and the additive group (픽p, +). We demonstrate that these codes are effective in detecting burst errors caused by hardware damage or noisy transmission. We also explore the codes ability to detect various types of double errors, including transposition errors, twin errors, and phonetic errors. Our analysis extends beyond bit-level errors to include block-level errors. Finally, we provide a comparative analysis and demonstrate the real-world applications of our code. [ABSTRACT FROM AUTHOR]

Codes over integer rings equipped with the Lee metric are gaining attention among researchers recently. In this paper, we investigate some specific linear codes over integer rings and characterize their dimensions and Lee weights using greatest common divisors (GCD). Furthermore, we construct equidistant Lee codes over ℤ2b, ℤ2pandℤp which have applications in error detection and error correction. [ABSTRACT FROM AUTHOR]

The construction of (minimal) linear codes from functions has received much attention in the literature. In this paper, we derive several minimal codes from the sets of pre-images of weakly regular plateaued and bent functions over the odd characteristic finite fields. Based on the recent construction method, we obtain six-weight and seven-weight minimal codes from these functions. The Hamming weights in proposed codes are calculated from the character sums of the sets based on the Walsh spectrum of the employed functions, while the weight distributions of the codes are determined from both the sizes of the employed sets and the Walsh distributions of the employed functions. [ABSTRACT FROM AUTHOR]

The primary application of codes with locality and availability in distributed storage is for data recovery in case that data are lost on some damaged servers. Locality enables the recovery of lost data by contacting only a restricted number of remaining servers. Availability means that more than one subset of servers providing locality is available for data recovery for each server. The secondary application of these codes in distributed storage is to allow access to hot data in times of high demand. In this paper it is shown that the binary linear [14, 4, 7] code has locality 2 and availability 6 and it can be interpreted as a three-dimensional graph obtained from a [7, 3, 4] Simplex code. It is achieving upper bounds on basic parameters for codes with all-symbols locality and availability. This code can be a building element of more complex codes with scalability inspired by three-dimensional structures. The availability spectrum is introduced as a tool for analyzing codes with locality and availability. [ABSTRACT FROM AUTHOR]

This paper provides the results of some computer searches for disjoint maximal (52, 4)-arcs in the known planes of order 16. Thirty-seven new such sets are discovered: four in Johnson plane and thirty-three in Mathon plane, eighteen of which give examples of 104-sets of type (4,8) coming from non-isomorphic pairs of maximal (52, 4)-arcs, providing first examples for such sets. A new lower bound on the number of 104-sets of type (4,8) coming from disjoint maximal (52, 4)-arcs in the known planes of order 16 is obtained. The 104-length binary and ternary linear codes generated by the blocks of 1-designs associated with the known 104-sets of type (4,8) are classified. [ABSTRACT FROM AUTHOR]

Linear complementary dual codes (LCD codes for short) are an important subclass of linear codes which have nice applications in communication systems, cryptography, consumer electronics and information protection. In the literature, it has been proved that an [ n , k , d ] Euclidean LCD code over F q with q > 3 exists if there is an [ n , k , d ] linear code over F q , where q is a prime power. However, the existence of binary and ternary Euclidean LCD codes has not been totally characterized. Hence it is interesting to construct binary and ternary Euclidean LCD codes with new parameters. In this paper, we construct new families of binary and ternary leading-systematic Euclidean LCD codes from some special functions including semibent functions, quadratic functions, almost bent functions, and planar functions. These LCD codes are not constructed directly from such functions, but come from some self-orthogonal codes constructed with such functions. Compared with known binary and ternary LCD codes, the LCD codes in this paper have new parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study mixed alphabet codes over two alphabets Z2, and its extension Z2[u] = Z2 + uZ2, u2 = 0. We define new Gray maps, using which we study cyclic codes and obtain the properties of their generators. Further, we construct constacyclic codes, and linear codes, and observe the useful properties of their generators. Finally, we present some examples of cyclic codes and binary linear codes. [ABSTRACT FROM AUTHOR]

The classical way of extending an [ n , k , d ] linear code C is to add an overall parity-check coordinate to each codeword of the linear code C. This extended code, denoted by C ‾ (− 1) and called the standardly extended code of C , is a linear code with parameters [ n + 1 , k , d ¯ ] , where d ¯ = d or d ¯ = d + 1. 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 have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) codes, which are a class of linear codes introduced by Massey, have been extensively studied in the literature recently. It has been shown that LCD codes play a role in measures to counter passive and active side-channel analyses on embedded cryptosystems. In this paper, tables are presented of good quaternary Hermitian LCD codes and they are used in the construction of puncturing, shortening and combination codes. The results of this, including three tables of the best-known quaternary Hermitian LCD codes of any length n≤25 with corresponding dimension k, are presented. In addition, many of these quaternary Hermitian LCD codes given in this paper are optimal and saturate the lower or upper bound of Grassl’s code table, and some of them are nearly optimal.

There are many infinite families of 2-designs and 3-designs supported by linear codes in the literature. Recently, several infinite families of 4-designs from linear codes were discovered. But no infinite family of linear codes supporting an infinite family of nontrivial simple 5-designs is reported in the literature. It is not easy to construct 5-designs from sporadic codes. Only a small number of sporadic 5-designs supported by linear codes are available in the literature. In this paper, we construct eight 5-designs from some lifted codes of some known linear codes. • There are many infinite families of 2-designs and 3-designs supported by linear codes in the literature. • Recently, several infinite families of 4-designs from linear codes were discovered. • But no infinite family of linear codes supporting an infinite family of nontrivial simple 5-designs is reported in the literature. • It is not easy to construct 5-designs from sporadic codes. Only a small number of sporadic 5-designs supported by linear codes are available in the literature. • In this paper, we construct eight new 5-designs from some lifted codes of some known linear codes. [ABSTRACT FROM AUTHOR]

In this article we mainly study linear codes over F 2 n and their binary subfield codes. We construct linear codes over F 2 n whose defining sets are the certain subsets of F 2 n m obtained from mathematical objects called simplicial complexes. We will consider simplicial complexes with one maximal element. The relation of the weights of codewords in two special codes obtained from simplicial complexes is illustrated by using LFSR sequences. And then we determine the parameters of these codes with the help of Boolean functions. As a result, we obtain five infinite families of distance optimal codes and give sufficient conditions for these codes to be minimal. [ABSTRACT FROM AUTHOR]

We study (1 + 2u + 2v)-constacyclic code over a semi-local ring S = Z4 + uZ4 + vZ4 with the condition u² = 3u, v² = 3v, and uv = vu = 0, we show that (1+2u+2v)-constacyclic code over S is equivalent to quasi-cyclic code over Z4 by using two new Gray maps from S to Z4. Also, for odd length n we have defined a generating set for constacyclic codes over S. Finally, we obtained some examples which are new to the data base [Database of Z4 codes [online], http://Z4 Codes.info(Accessed March 2, 2020)]. [ABSTRACT FROM AUTHOR]

There are perfect and ideal threshold secret sharing schemes, for example, Shamir’s secret sharing scheme. For the case of general secret sharing schemes with an arbitrary access structure, it is possible to construct a perfect scheme for any access structure (for example, the Ito – Saito – Nishizeki scheme, the Benaloh – Leichter scheme), but in general, such a scheme will not be an ideal secret sharing scheme. In the paper, for some classes of access structures, the construction of perfect and ideal secret sharing schemes based on linear codes is given. We also give a construction of perfect verifiable secret sharing schemes for any access structure for which there is a line code that implements this structure.

Reversible codes have a range of wide applications in cryptography, data storage, and communication systems. In this paper, we investigated reversible codes under the Rosenbloom-Tsfasman metric (RT-metric). First, some properties of reversible codes in the RT-metric were described. An essential condition for a reversible code to be a maximum distance separable code (MDS code, in short) in the RT-metric was established. A necessary condition for a binary self-dual code to be reversible was proven and the same was generalized for $ q $-ary self-dual reversible codes. Several constructions for reversible RT-metric codes were provided in terms of their generator matrices.

We introduce a new class of linear codes over a finite field associated to determinantal varieties in the space of hermitian matrices and determine their length, dimension and minimum distance along with the weight spectrum. [ABSTRACT FROM AUTHOR]

Linear codes are the most important family of codes in cryptography and coding theory. Some codes only have a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting p ≡ 1 (mod 4) , we constructed an infinite family of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions with index (p − 1) / 2 . Their weight distributions were completely determined by applying exponential sums and Walsh transform. As a result, most of our constructed codes have a few nonzero weights and are minimal. [ABSTRACT FROM AUTHOR]

The article titled "THREE-WEIGHT AND FIVE-WEIGHT LINEAR CODES OVER FINITE FIELDS" published in the Kragujevac Journal of Mathematics discusses the construction and properties of linear codes derived from defining sets. The authors define a pe-ary linear code C_D using a defining set D and present three-weight and five-weight linear codes along with their weight distributions. They demonstrate that these codes are minimal for m≥5 and can be applied in secret sharing schemes. The article also includes preliminary concepts such as cyclotomic numbers and Gauss sums, which are used in the construction of the linear codes. [Extracted from the article]

Let p be an odd prime number, q = p m for a positive integer m , let q be the finite field with q elements and ω be a primitive element of q . We first give an orthogonal decomposition of the ring R = q + ν q , where ν 2 = a 3 , and a = ω 2 l for a fixed integer l. In addition, Galois dual of a linear code over R is discussed. Meanwhile, constacyclic codes and cyclic codes over the ring R are investigated as well. Remarkably, we obtain that if linear codes and are a complementary pair, then the code and the dual code ⊥ E of are equivalent to each other. [ABSTRACT FROM AUTHOR]

Recently, the research of trace codes over finite fields has received a lot of attention because some of these codes have good parameters. In this paper, we construct several classes of trace codes from some special classes of linear codes and give their weight distributions, showing that they contain some optimal and almost optimal codes. Furthermore, we determine the parameters of the duals of the trace codes, and show that they are distance-optimal or nearly distance-optimal with respect to the sphere-packing bound. [ABSTRACT FROM AUTHOR]

An algorithm for equivalence of linear codes over finite fields is presented. Its main advantage is that it can extract exactly one representative from each equivalence class among a large number of linear codes. It can also be used as a test for isomorphism of binary matrices. The algorithm is implemented in the program LCequivalence, which is designed to obtain the inequivalent codes in a set of linear codes over a finite field with q < 64 elements. This program is a module of the free software package QextNewEdition for constructing, classifying and studying linear codes. [ABSTRACT FROM AUTHOR]

In this paper, three discrete transforms related to vector spaces over finite fields are studied. For our purposes, and according to the properties of the finite fields, the most suitable transforms are as follows: for binary fields, this is the Walsh–Hadamard transform; for odd prime fields, the Vilenkin–Chrestenson transform; and for composite fields, the trace transform. A factorization of the transform matrices using Kronecker power is given so that the considered discrete transforms are reduced to the fast discrete transforms. Examples and applications are also presented of the considered transforms in coding theory for calculating the weight distribution of a linear code.