A Class of Affine-Invariant Codes and Their Related Codes
Chengju Li, Chunyu Gan · SIAM Journal on Discrete Mathematics · 2025
Abstract. Affine-invariant codes are an important class of linear codes, which are extended cyclic codes of length [Formula: see text] invariant under the affine groups acting on [Formula: see text]. These codes are closely related to combinatorics, as they can be applied to construct [Formula: see text]-designs. It is known that the classical Reed–Muller codes and extended primitive narrow-sense Bose–Chaudhuri–Hocquenghem codes are affine-invariant. The objective of this paper is to construct a class of affine-invariant codes [Formula: see text] and investigate the parameters of these codes and their related codes. The dimensions of the codes [Formula: see text] and [Formula: see text] with [Formula: see text] are presented and a recursive formula to compute the dimensions of [Formula: see text] and [Formula: see text] is developed for general [Formula: see text], where [Formula: see text] is the extended code of [Formula: see text]. Meanwhile, lower bounds on minimum distances of [Formula: see text] and [Formula: see text] are also given. Moreover, the parameters and the borders of the dual codes [Formula: see text] are investigated. Two necessary and sufficient conditions for [Formula: see text] being self-orthogonal with [Formula: see text] are developed by employing their borders. In addition, for [Formula: see text] we explore the parameters of the hull of [Formula: see text] and determine its border. It should be pointed out that several affine-invariant self-orthogonal codes will be obtained.