Binary Linear Codes With Two Weights

Abstract

Linear codes have wide applications in secret sharing schemes and authentication codes. In this letter, a class of binary linear codes with two weights is presented. The weight distribution of this class of binary linear codes is settled. The dual codes are also studied and proved to be optimal or almost optimal.

Complete weight enumerators of two classes of linear codes

“…This construction technique is employed in lots of researches to get linear codes with few weights. The readers are referred to [17], [29], [35], [12], [24], [29] for more details. Naturally, a generalization of the code C D of (1.1) is defined by [32] …”

Complete weight enumerators of two classes of linear codes with a few weights

“…This section is devoted to prove the main theorems of the previous section, before that some concepts and basic results on group character and a series of needed lemmas, such as the work of Coulter on exponential sum over finite field [8,9], some intermediate results in [6], will be presented. At first, we introduce some sums of Legendre's Symbols needed in the estimation of the Hamming weights from the corresponding complete weight enumerators.…”

“…It is clear that n a of (14) gives the length of codewords of the linear codes C D a defined in (5) and constructed by using the defining set of (13), whose values were explicitly computed in [6], resumed by the following lemma:…”

“…[6],Lemma 30). Suppose that m/d is even, and a ∈ F * p .For b ∈ F * q , if f (X) = X p 2α + X = −b p α has no solution in F q , then |N b (a, 0)| = p e−2 + p m+d−2 , else |N b (a, 0)…”

A construction of codes with linearity from two linear codes

“…We refer the reader to [17,20,23,31,34] for other linear codes with a few weights or optimal parameters.…”