Infinite families of 2‐designs derived from affine‐invariant codes

Abstract

In this paper, we first study a general type of affine‐invariant codes and their support 2‐designs. Then we derive infinite families of 2‐designs from a special class of affine‐invariant codes and obtain their parameters by determining the weight distribution of the linear codes.

Infinite families of 2-designs from a class of affine-invariant codes

“…In this paper, we give a coding theoretical construction of infinite families of 2designs. It should be noticed that the linear codes C ⊥ considered in this paper can be viewed as a generalization of the affine-invariant codes considered in [18]. Besides, the parameters of their support 2-designs depend on the weight distributions of the linear codes.…”

“…It should be noticed that 2, p k + 1 and p 2k + 1 are in different cyclotomic cosets from [17]. Then C can be viewed as a special case of the general type of cyclic codes considered in [18]. Then by the results obtained in [18], the dimension of C is p m − 1 − 4m and C…”

“…Then C can be viewed as a special case of the general type of cyclic codes considered in [18]. Then by the results obtained in [18], the dimension of C is p m − 1 − 4m and C…”

“…⊥ is affine-invariant. Besides, in [18], we consider the code in the case of gcd(m, k) = 1. Then we generalize the result about this code into a general case in this paper.…”