Family Of Binary Linear Codes Research Articles

On Some Families of Codes Related to the Even Linear Codes Meeting the Grey–Rankin Bound

Bounds for the parameters of codes are very important in coding theory. The Grey–Rankin bound refers to the cardinality of a self-complementary binary code. Codes meeting this bound are associated with families of two-weight codes and other combinatorial structures. We study the relations among six infinite families of binary linear codes with two and three nonzero weights that are closely connected to the self-complementary linear codes meeting the Grey–Rankin bound. We give a construction method and partial classification results for such codes. The properties of the codes in the studied families and their relations help us in constructing codes of a higher dimension from codes with a given dimension.

Some Punctured Codes of Several Families of Binary Linear Codes

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(f)={ Tr(af(x)+bx)x ∈ \mathbb Fqm*: a,b ∈ \mathbb Fqm }, where q is a prime power, \mathbb Fqm* = \mathbb Fqm \{0}, Tr is the trace function from \mathbb Fqm to \mathbb Fq, and f(x) is a function from \mathbb Fqm to \mathbb Fqm with f(0)=0. Almost bent functions, quadratic functions and some monomials on \mathbb 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.

Binary linear codes with few weights from Boolean functions

Boolean functions have very nice applications in coding theory and cryptography. In coding theory, Boolean functions have been used to construct linear codes in different ways. The objective of this paper is to construct binary linear codes with few weights using the defining-set approach. The defining sets of the codes presented in this paper are defined by some special Boolean functions and some additional restrictions. First, two families of binary linear codes with at most three or four weights from Boolean functions with at most three Walsh transform values are constructed and the parameters of their duals are also determined. Then several classes of binary linear codes with explicit weight enumerators are produced. Some of the binary linear codes are optimal or almost optimal according to the tables of best codes known maintained at http://www.codetables.de , and the duals of some of them are distance-optimal with respect to the sphere packing bound.

Projective binary linear codes from special Boolean functions

Linear codes with a few weights have nice applications in communication, secret sharing schemes, authentication codes, association schemes, block designs and so on. Projective binary linear codes are one of the most important subclasses of linear codes for practical applications. The objective of this paper is to construct projective binary linear codes with some special Boolean functions. Four families of binary linear codes with three or four weights are derived and the parameters of their duals are also determined. It turns out that the duals of these codes are optimal or almost optimal with respect to the sphere-packing bound. As applications, the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures.

Bent Vectorial Functions, Codes and Designs

Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group ( ${\mathrm {GF}}(2^{2m}), $ +), have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus–Mattson theorem, but nevertheless hold 2-designs. A new coding-theoretic characterization of bent vectorial functions is presented.

Strong Locally Testable Codes with Relaxed Local Decoders

Locally testable codes (LTCs) are error-correcting codes that admit very efficient codeword tests. An LTC is said to be strong if it has a proximity-oblivious tester, that is, a tester that makes only a constant number of queries and rejects non-codewords with a probability that depends solely on their distance from the code. Locally decodable codes (LDCs) are complementary to LTCs. While the latter allow for highly efficient rejection of strings that are far from being codewords, LDCs allow for highly efficient recovery of individual bits of the information that is encoded in strings that are close to being codewords. Constructions of strong-LTCs with nearly-linear length are known, but the existence of a constant-query LDC with polynomial length is a major open problem. In an attempt to bypass this barrier, Ben-Sasson et al. (SICOMP 2006) introduced a natural relaxation of local decodability, called relaxed-LDCs. This notion requires local recovery of nearly all individual information-bits, yet allows for recovery-failure (but not error) on the rest. Ben-Sasson et al. constructed a constant-query relaxed-LDC with nearly-linear length (i.e., length k 1+α for an arbitrarily small constant α > 0, where k is the dimension of the code). This work focuses on obtaining strong testability and relaxed decodability simultaneously . We construct a family of binary linear codes of nearly-linear length that are both strong-LTCs (with one-sided error) and constant-query relaxed-LDCs. This improves upon the previously known constructions, which either obtain only weak LTCs or require polynomial length. Our construction heavily relies on tensor codes and PCPs. In particular, we provide strong canonical PCPs of proximity for membership in any linear code with constant rate and relative distance. Loosely speaking, these are PCPs of proximity wherein the verifier is proximity oblivious (similarly to strong-LTCs) and every valid statement has a unique canonical proof. Furthermore, the verifier is required to reject non-canonical proofs (even for valid statements). As an application, we improve the best known separation result between the complexity of decision and verification in the setting of property testing.

Four families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ w min / w max ≤ 1 / 2

As a special type of linear codes, minimal linear codes have important applications in secret sharing. Up to now, only a few infinite families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ were reported in the literature, while vast knowledge exists on the ones with $$w_{\min }/w_{\max }> 1/2$$ . Herein, $$w_{\min }$$ and $$w_{\max }$$ respectively denote the minimum and maximum nonzero Hamming weights in a linear code. Recently, several classes of linear codes with certain properties were constructed by Zhou et al. from a generic construction. The objective of this paper is to obtain four families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ from those linear codes proposed by Zhou et al. The parameters of our minimal linear codes are quite different from known ones. Based on the properties of Krawtchouk polynomials, the weight distributions of all these four families of binary linear codes are established.

Infinite families of recursive formulas generating power moments of Kloosterman sums: O −(2n,2r) case

Abstract In this paper, we construct eight infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the special orthogonal group SO −(2n, 2r). And we obtain four infinite families of recursive formulas for the power moments of Kloosterman sums and four those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of “Gauss sums” for the orthogonal groups O −(2n, 2r).

A Family of Binary (t, m,s)-Nets of Strength 5

(t,m,s)-Nets were defined by Niederreiter [Monatshefte fur Mathematik, Vol. 104 (1987) pp. 273–337], based on earlier work by Sobol’ [Zh. Vychisl Mat. i mat. Fiz, Vol. 7 (1967) pp. 784–802], in the context of quasi-Monte Carlo methods of numerical integration. Formulated in combinatorial/coding theoretic terms a binary linear (m−k,m,s)2-net is a family of ks vectors in F m2 satisfying certain linear independence conditions (s is the length, m the dimension and k the strength: certain subsets of k vectors must be linearly independent). Helleseth et al. [5] recently constructed (2r−3,2r+2,2r−1)2-nets for every r. In this paper, we give a direct and elementary construction for (2r−3,2r+2,2r+1)2-nets based on a family of binary linear codes of minimum distance 6.