Minimal Linear Codes Research Articles

Strong blocking sets and minimal codes from expander graphs

A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the ( k − 1 ) (k-1) -dimensional projective space over F q \mathbb {F}_q that have size at most c q k c q k for some universal constant c c . Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of F q \mathbb {F}_q -linear minimal codes of length n n and dimension k k , for every prime power q q , for which n ≤ c q k n \leq c q k . This solves one of the main open problems on minimal codes.

Infinite families of minimal binary codes via Krawtchouk polynomials

Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, namely, the general Maiorana–McFarland class. We employ a method first proposed by Ding et al. (IEEE Trans Inf Theory 64(10):6536–6545, 2018) to construct minimal codes violating the Ashikhmin–Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. These general results offer insights into the scope of this approach.

Minimal Linear Codes Constructed from Sunflowers.

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 Fq 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.

Minimal linear codes constructed from partial spreads

Minimal linear codes constructed from partial spreads

Construction of Two Classes of Minimal Binary Linear Codes from Definition Sets

Construction of Two Classes of Minimal Binary Linear Codes from Definition Sets

Minimal linear codes derived from weakly regular bent and plateaued functions

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.

Minimal linear codes from defining sets over [formula omitted

Minimal codes are a special type of linear codes, which have nice applications in secret sharing and secure two-party computation. How to construct new infinite family of minimal codes has been a hot topic in coding theory and cryptography. In this paper, by employing exponential sums, we study the Lee-weight distributions of several classes of linear codes from defining sets over the finite chain ring Fp+uFp and determine exactly the complete weight enumerator of their Gray images under the Gray map. Furthermore, we prove that the Gray images of these linear codes are new infinite families of minimal five-weight linear codes with wminwmax<p−1p.

Minimal Binary Linear Codes From Vectorial Boolean Functions

Recently, much progress has been made to construct minimal linear codes due to their preference in secret sharing schemes and secure two-party computation. In this paper, we put forward a new method to construct minimal linear codes by using vectorial Boolean functions. Firstly, we give a necessary and sufficient condition for a generic class of linear codes from vectorial Boolean functions to be minimal. Based on that, we derive some new three-weight minimal linear codes and determine their weight distributions. Secondly, by studying deeply the construction of linear codes in this paper, we find a necessary and sufficient condition of the linear codes to be minimal and to be violated the AB condition. As a result, we get three infinite families of minimal linear codes violating the AB condition. To the best of our knowledge, this is the first time that minimal liner codes are constructed from vectorial Boolean functions. Compared the parameters with other known ones, in general the minimal liner codes obtained in this paper have higher dimensions.

Several Families of Binary Minimal Linear Codes From Two-to-One Functions

Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. K. Li, C. Li, T. Helleseth, and L. Qu have recently identified in [Binary linear codes with few weights from two-to-one functions, IEEE Trans. Inf. Theory, 2021] some binary linear codes with few weights from two classes of two-to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> . by proposing larger classes of binary linear codes with few weights via generic constructions involving other known families of two-to-one functions over the finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^<i>n</i></sub> of order 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sup> . We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with respect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> . (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).

Weight distributions of generalized quasi-cyclic codes over [formula omitted

In this paper, by Gauss sums, we determine Hamming weight distributions of generalized quasi-cyclic (GQC) codes over the finite chain ring Fq+uFq, where u2=0. As applications, some new infinite families of minimal linear codes with WminWmax≤q−1q and projective two-weight codes are constructed.

Minimal linear codes constructed from hierarchical posets with two levels

Minimal linear codes constructed from hierarchical posets with two levels

Minimal binary linear codes: a general framework based on bent concatenation

Minimal binary linear codes: a general framework based on bent concatenation

Construction of Two Classes of Minimal Binary Linear Codes Based on Boolean Function

Construction of Two Classes of Minimal Binary Linear Codes Based on Boolean Function

Three-Weight Minimal Linear Codes

Three-Weight Minimal Linear Codes

Short Minimal Codes and Covering Codes via Strong Blocking Sets in Projective Spaces

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. Minimal linear codes have been studied since decades but their tight connection with cutting blocking sets of finite projective spaces was unfolded only in the past few years, and it has not been fully exploited yet. In this paper we apply finite geometric and probabilistic arguments to contribute to the field of minimal codes. We prove an upper bound on the minimal length of minimal codes of dimension <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> over the <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-element Galois field which is linear in both <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well.

Minimal linear codes constructed from functions

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.

On cutting blocking sets and their codes

AbstractLetPG⁡(r,q){\operatorname{PG}(r,q)}be ther-dimensional projective space over the finite fieldGF⁡(q){\operatorname{GF}(q)}. A set𝒳{\mathcal{X}}of points ofPG⁡(r,q){\operatorname{PG}(r,q)}is a cutting blocking set if for each hyperplane Π ofPG⁡(r,q){\operatorname{PG}(r,q)}the setΠ∩𝒳{\Pi\cap\mathcal{X}}spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set ofPG⁡(3,q3){\operatorname{PG}(3,q^{3})}of size3⁢(q+1)⁢(q2+1){3(q+1)(q^{2}+1)}as a union of three pairwise disjointq-order subgeometries, and a cutting blocking set ofPG⁡(5,q){\operatorname{PG}(5,q)}of size7⁢(q+1){7(q+1)}from seven lines of a Desarguesian line spread ofPG⁡(5,q){\operatorname{PG}(5,q)}. In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimalq-ary linear code having dimension 4 and 6.

Construction of minimal linear codes with few weights from weakly regular plateaued functions

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.

New classes of few-weight ternary codes from simplicial complexes

&lt;abstract&gt;&lt;p&gt;In this article, we describe two classes of few-weight ternary codes, compute their minimum weight and weight distribution from mathematical objects called simplicial complexes. One class of codes described here has the same parameters with the binary first-order Reed-Muller codes. A class of (optimal) minimal linear codes is also obtained in this correspondence.&lt;/p&gt;&lt;/abstract&gt;

Minimal linear codes from weakly regular bent functions

Minimal linear codes from weakly regular bent functions