Some punctured codes of several families of binary linear codes

Abstract

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∈F * q m : a, b ∈ F q m , where q is a prime power, F * q m = F q m \ {0}, Tr is the trace function from F q m to F q , and f (x) is a function from F q m to F q m with f (0) = 0. Almost bent functions, quadratic functions and some monomials on F 2 m 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.

Primitive Rateless Codes

“…and has weight t. There are n − c of such multiples, where max{k, t − 1} ≤ c ≤ n − 1. Therefore, the expected number of weight t polynomials of maximum degree n − 1, which are multiples of primitive polynomial p(x), is given by (13).…”

“…codes [1], [4], convolutional codes [5], [6], Turbo codes [7], [8], low-density parity-check (LDPC) codes [9], [10], and Polar codes [11], [12]. The performance of the resulting code depends mainly on the puncturing pattern [13]. Finding the best puncturing pattern is usually nontrivial and carried out through computer search.…”