Codes with few weights arising from linear sets (original) (raw)

The Weight Enumerator for a Class of Linear Codes

Recently, linear codes constructed from defining sets have been studied widely since they have many applications in cryptography and communication systems. In this paper, we consider a defining set where for a positive integer and an odd prime , and is the absolute trace function from onto. Define a class of-ary linear codes by where We compute the weight enumerators of the punctured codes .

Linear authentication codes: bounds and constructions

IEEE Transactions on Information Theory, 2003

In this paper, we consider a new class of unconditionally secure authentication codes, called linear authentication codes (or linear A-codes). We show that a linear A-code can be characterized by a family of subspaces of a vector space over a finite field. We then derive an upper bound on the size of source space when other parameters of the system, that is, the sizes of the key space and the authenticator space, and the deception probability, are fixed. We give constructions that are asymptotically close to the bound and show applications of these codes in constructing distributed authentication systems.

Generalized Hamming weights of three classes of linear codes

Finite Fields and Their Applications, 2017

The generalized Hamming weights of a linear code have been extensively studied since Wei first use them to characterize the cryptography performance of a linear code over the wire-tap channel of type II. In this paper, we investigate the generalized Hamming weights of three classes of linear codes constructed through defining sets and determine them partly for some cases. Particularly, in the semiprimitive case we solve an problem left in Yang et al. (2015) [30]. Keywords Linear codes • Generalized Hamming weights • Gauss periods 1 Introduction Throughout this paper let p be an odd prime and q = p m for some positive integer m. Let n 1 be a positive integer coprime to p and without loss of generality we assume that m is the least positive integer such that p m ≡ 1 (mod n 1). Denote by Fp (or Fq) the finite field with p (or q) elements. Let α be a fixed primitive element of Fq. Let N = q−1 n1 , N 1 = gcd(N, q−1 p−1), N 2 = lcm(N, q−1 p−1) and θ = α N. Let Tr denote the trace function from Fq to Fp. An [n, k, d] linear code C over Fp is a k-dimensional subspace of F n p with minimum distance d. We recall the definition of the generalized Hamming weights of a linear code [28]. Suppose that U is an r-dimensional subspace of C, the support of U is defined to be Supp(U) = ∪ x∈U Supp(x), where Supp(x) is the set of coordinates where x is nonzero, i.e., Supp(U) = {i : 1 ≤ i ≤ n, x i = 0 for some x = (x 1 , x 2 ,. .. , xn) ∈ U }. Definition 1 Let C be an [n, k, d] linear code over Fp. For 1 ≤ r ≤ k, dr(C) = min{|Supp(U)| : U ⊂ C, dim U = r} is called the r-th generalized Hamming weight (GHW) of C and {dr(C) : 1 ≤ r ≤ k} is called the weight hierarchy of C.

Optimal non-projective linear codes constructed from down-sets

Discrete Applied Mathematics, 2019

By an optimal linear code we mean that it has the highest minimum distance with a prescribed length and dimension. We construct several families of optimal linear codes over the finite field F p by making use of down-sets generated by one maximal element of F n p. Moreover, we show that these families of optimal linear codes are minimal and contain relative two-weight linear codes, and have applications to secret sharing schemes and wire-tap channel of type II with the coset coding scheme, respectively.

On relative constant-weight codes

In this paper, relative two-weight and three-weight codes are studied, which are both called relative constant-weight codes. A geometric approach is introduced to construct and characterize relative constant-weight codes, using the finite projective geometry.

Three-weight codes and the quintic construction

ArXiv, 2016

We construct a class of three-Lee-weight and two infinite families of five-Lee-weight codes over the ring R=mathbbF2+vmathbbF2+v2mathbbF2+v3mathbbF2+v4mathbbF_2,R=\mathbb{F}_2 +v\mathbb{F}_2 +v^2\mathbb{F}_2 +v^3\mathbb{F}_2 +v^4\mathbb{F}_2,R=mathbbF_2+vmathbbF_2+v2mathbbF_2+v3mathbbF_2+v4mathbbF_2, where v5=1.v^5=1.v5=1. The same ring occurs in the quintic construction of binary quasi-cyclic codes. %The length of these codes depends on the degree mmm of ring extension. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Given a linear Gray map, we obtain three families of binary abelian codes with few weights. In particular, we obtain a class of three-weight codes which are optimal. Finally, an application to secret sharing schemes is given.

Several classes of minimal binary linear codes violating the Ashikhmin-Barg bound

Cryptography and Communications, 2021

Minimal linear codes are a special class of codes which have important applications in secret sharing and secure two-party computation. These codes are characterized by the property that none of the codewords is covered by some other codeword. Denoting by w min and w max minimal and maximal weight of the codewords respectively, such codes are relatively easy to design when the ratio w min /w max > 1/2 (known as Aschikhmin-Barg's bound). On the other hand, there are few known classes of minimal codes violating this bound, hence having the property w min /w max ≤ 1/2. In this article, we provide several explicit classes of minimal binary linear codes violating the Aschikhmin-Barg's bound, at the same time achieving a great variety of the ratio w min /w max. Our first generic method employs suitable characteristic functions of relatively low weight within the range [n + 1, 2 n−2 ]. The second approach addresses a specification of characteristic functions covering the weights in [2 n−2 + 1, 2 n−2 + 2 n−3 − 1] and containing a skewed (removing one element) affine subspace of dimension n − 2. Finally, we also characterize an infinite family of such codes that utilize the class of so-called root Boolean functions of weight 2 n−1 − (n − 1), which are useful in certain hardware testing applications. Consequently, many infinite classes of minimal codes crossing the Aschikhmin-Barg's bound, with a wide range of the weight of their characteristic functions, are deduced. In certain cases we also completely specify the weight distribution of resulting codes.

Construction of Two- or Three-Weight Binary Linear Codes from Vasil'Ev Codes

Journal of The Korean Mathematical Society, 2021

The set D of column vectors of a generator matrix of a linear code is called a defining set of the linear code. In this paper we consider the problem of constructing few-weight (mainly twoor three-weight) linear codes from defining sets. It can be easily seen that we obtain an oneweight code when we take a defining set to be the nonzero codewords of a linear code. Therefore we have to choose a defining set from a non-linear code to obtain twoor three-weight codes, and we face the problem that the constructed code contains many weights. To overcome this difficulty, we employ the linear codes of the following form: Let D be a subset of F2 , and W (resp. V ) be a subspace of F2 (resp. F2 ). We define the linear code CD(W ;V ) with defining set D and restricted to W,V by CD(W ;V ) = {(s + u · x)x∈D∗ | s ∈W,u ∈ V }. We obtain twoor three-weight codes by taking D to be a Vasil’ev code of length n = 2m − 1(m ≥ 3) and a suitable choices of W . We do the same job for D being the complement o...

Three constructions of authentication/secrecy codes

Journal of Pure and Applied Algebra, 2005

In this paper, we present three algebraic constructions of authentication codes with secrecy. The codes have simple algebraic structures and are easy to implement. They are asymptotically optimal with respect to certain bounds.