Projective Linear Codes From Some Almost Difference Sets

Abstract

Projective linear codes are a special class of linear codes whose duals have minimum distance at least 3. The columns of the generator matrix of an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n,k]$ </tex-math></inline-formula> projective code over finite field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> can be viewed as points in the projective space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text {PG}(k-1, {\mathbb {F}}_{q})$ </tex-math></inline-formula> . Projective codes are of interest not only because their duals have good error correcting capability but also because they may be related to interesting combinatorial structures. The objective of this paper is to construct projective linear codes with five families of almost difference sets. To this end, the augmentation and extension techniques for linear codes are used. The parameters and weight distributions of the projective codes are explicitly determined. Several infinite families of optimal or almost optimal codes including MDS codes, near MDS codes, almost MDS odes and Griesmer codes are obtained. Besides, we also give some applications of these codes.