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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]

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]

In [C. Ding, An infinite family of Steiner systems $ S(2, 4, 2^m) $ from cyclic codes, J. Combin. Des. 26 (2018), no.3, 126–144], Ding constructed a family of Steiner systems $ S(2, 4, 2^m) $ for all $ m \equiv 2 \pmod{4} \ge 6 $ from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems $ S(2, 4, 2^m) $ for all $ m \equiv 0 \pmod{4} \ge 4 $ supported by this family of extended cyclic codes. The main result of this paper complements the previous work of Ding, and the results in the two papers will show that there exists a binary extended cyclic code that can support a Steiner system $ S(2, 4, 2^m) $ for all even $ m \geq 4 $. Furthermore, this paper also determines the parameters of other $ 2 $-designs supported by this family of extended cyclic codes. [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]

Separating codes have been studied due to their applications to digital finger printing, the state assignments, automata theory and to construct hash functions. In this paper, we study the necessary and sufficient conditions for a code to be a (3 , 1) and (3 , 2) -separating systems for q-ary level and also satisfy its intersecting properties. We also construct a bound for (3 , 1) separating system. [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]

Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, a class of binary subfield codes is constructed from a special family of MDS codes, and their parameters are explicitly determined. The parameters of their dual codes are also studied. Some of the codes presented in this paper are optimal or almost optimal. [ABSTRACT FROM AUTHOR]

Let p ≡ 1 (mod 4) be a prime, m a positive integer, ϕ (p m) 2 the multiplicative order of 2 modulo p m , and let q = 2 ϕ (p m) 2 , where ϕ (⋅) is the Euler's function. In this paper, we construct two classes of linear codes over F q and investigate their weight distributions. By calculating two classes of special exponential sums, the desired results are obtained. Let be a prime, a positive integer, the multiplicative order of modulo , and let , where is the Euler's function. In this paper, we construct two classes of linear codes over and investigate their weight distributions. By calculating two classes of special exponential sums, the desired results are obtained. [ABSTRACT FROM AUTHOR]

Sunflower in coding theory is a class of important subspace codes and can be used to construct linear codes. In this paper, we study the minimality of linear codes over F q constructed from sunflowers of size s in all cases. For any sunflower, the corresponding linear code is minimal if s ≥ q + 1 , and not minimal if 2 ≤ s ≤ 3 ≤ q . In the case where 3 < s ≤ q , for some sunflowers, the corresponding linear codes are minimal, whereas for some other sunflowers, the corresponding linear codes are not minimal. [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing minimal binary codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity, are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension $m$ and length less than or equal to $m(m+1)/2$ are obtained. Besides, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields additionally to symmetric cryptography, such as coding theory and secret sharing schemes. [ABSTRACT FROM AUTHOR]

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(ƒ) = {Tr(aƒ(x) + bx)x∈Fqm*: a, b ∈ Fqm}, where q is a prime power, Fqm* = Fqm \ {0}, Tr is the trace function from Fqm to Fq, and ƒ(x) is a function from Fqm to Fqm with ƒ(0) = 0. Almost bent functions, quadratic functions and some monomials on F2m were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper. [ABSTRACT FROM AUTHOR]

The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes over finite fields are presented in this paper. As a byproduct, five infinite families of 2-designs are also constructed from some of the shortened codes presented in this paper. [ABSTRACT FROM AUTHOR]

The generalized Hamming weights of linear codes have attracted scholars' attention since Wei used them to characterize the cryptography performance of a linear code over the wire-tap channel of type II in 1991. Generally speaking, it is hard to determine linear codes' generalized Hamming weights, especially the weight hierarchy. On the other hand, since Ding and Niederreiter presented a generic approach toconstruct linear codes with good properties by suitable defining sets in 2007, many linear codes with few weights have been obtained using cryptographic functions. However, there does not seem to be much research on the generalized Hamming weights of linear codes from cryptographic functions.In this paper, we first provide a general formula to compute the generalized Hamming weights of linear codes from the defining sets, which generalizes several known results. Next, some weight hierarchies of linear codes from three famous classes of Bent functions (the Maiorana-McFarland class, the class $ \mathcal{PS}_{ap} $, the class $ H $) and two classes of Semi-Bent functions (the Maiorana-McFarland class, the summation of $ \mathcal{PS}_{ap} $ and $ H $) are considered. We hope this paper can attract interested scholars to find more results about the generalized Hamming weights of linear codes from cryptographic functions. [ABSTRACT FROM AUTHOR]

Recently, a number of researches studied relationship between codes and BCK-algebras, dealing with the category codes in a BCK-algebra have not been considered in earlier works. This paper investigates a code constructed by a BCK-algebra and also a BCK-algebra constructed based on code. The suggested rendered algorithm constructs the code based on BCK-algebra, the fixed dimension of this code is guaranteed by a BCK-algebra with some condition. We introduce the category of linear codes and denoted by LC. In addition, a subcategory of BCK-algebra is described and denoted by E BCK . Finally, we prove that LC is equivalent to E BCK . [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]

In this paper, we consider minimal linear codes by a general construction of linear codes from q-ary functions. First, we give necessary and sufficient conditions for codewords which are constructed by functions to be minimal. Second, as applications, we present three constructions of minimal linear codes. Constructions on minimal linear codes in this paper generalize some recent results in Ding et al. (IEEE Trans. Inf. Theory 64(10), 6536–6545, 2018); Heng et al. (Finite Fields Appl. 54, 176–196, 2018); Bartoli and Bonini (IEEE Trans. Inf. Theory 65(7), 4152–4155, 2019); Mesnager et al. (IEEE Trans. Inf. Theory 66(9), 5404–5413, 2020); Bonini and Borello (J. Algebraic Comb. 53, 327–341, 2021). In our three constructions, the conditions of functions are much looser than theirs. [ABSTRACT FROM AUTHOR]

Recently, subfield codes of geometric codes over large finite fields ${\mathrm {GF}}(q)$ with dimension 3 and 4 were studied and distance-optimal subfield codes over ${\mathrm {GF}}(p)$ were obtained, where $q=p^{m}$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of $[q+1, 2, q]$ MDS codes over ${\mathrm {GF}}(q)$ , and then studies the subfield codes over ${\mathrm {GF}}(p)$ of some of the $[q+1, 2,q]$ MDS codes over ${\mathrm {GF}}(q)$. Two families of dimension-optimal codes over ${\mathrm {GF}}(p)$ are obtained, and several families of nearly optimal codes over ${\mathrm {GF}}(p)$ are produced. Several open problems are also proposed in this paper. [ABSTRACT FROM AUTHOR]

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$ -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$ -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes. [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, two classes of cyclic codes whose duals have two zeros are presented. The weight distributions of these cyclic codes are settled with the help of Gaussian periods. The duals of one class of cyclic codes are also studied. Some of the cyclic codes presented in this paper are optimal in the sense that they meet some bounds of linear codes. [ABSTRACT FROM AUTHOR]

The construction of (minimal) linear codes from functions over finite fields has been greatly studied in the literature, since determining the parameters of codes based on functions is rather easy due to the nice structures of functions. In this paper, we derive 3-weight and 4-weight linear codes from weakly regular plateaued functions in the recent construction method of linear codes over the odd characteristic finite fields. The Hamming weights and weight distributions for codes are, respectively, determined by using the Walsh transform values and Walsh distributions of the employed functions. We also derive projective 3-weight punctured codes with good parameters from the constructed codes. These punctured codes may be almost optimal due to the Griesmer bound, and they can be employed to design association schemes. We lastly show that all constructed codes are minimal, which approves that they can be employed to design high democratic secret sharing schemes. [ABSTRACT FROM AUTHOR]

In this paper, a family of $[{q}^{2}-1, 4, {q}^{2}-{q}-2]$ cyclic codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2}-1,{q}^{2}-5,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of cyclic codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of $[{q}^{2}, 4, {q}^{2}-{q}-1]$ linear codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2},{q}^{2}-4,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of linear codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Five infinite families of 2-designs are also constructed with three families of linear codes of this paper. [ABSTRACT FROM AUTHOR]

Information retrieval in associative memories was studied in a recent paper by Yaakobi and Bruck (2012). Associations between memory entries give us the t -neighbourhood of an entry. In their model, an information unit is retrieved from the memory with the aid of input clues, which are chosen from a reference set. In this paper, we consider the situation where the information unit is found unambiguously using the associated t -neighbourhoods of the input clues. A varying number of input clues are allowed, but a limit mu on the maximum number of them is imposed. Of course, we would like mu to be as small as possible. We consider the problem over the binary Hamming space \mathbb F^n and focus on the minimum of mu , denoted by $\nu (n;t)$ . Using linear reference sets, we show that $\nu (n;2) \leq 5$ for any $n \geq 9$ . We also give infinite families of reference sets, which provide good bounds on $\nu (n;t)$ for $t=3$ . In addition, efficient methods are given to obtain bounds on $\nu (n;t)$ for any $t$ from known reference sets. We also discuss the applications of this model to the Levenshtein’s sequence reconstruction problem and the sensor network monitoring. [ABSTRACT FROM AUTHOR]