A Structural Characterization of Cyclotomic Cosets with Applications to Affine-Invariant Codes and BCH Codes

TL;DR

This paper characterizes cyclotomic cosets and applies the findings to derive explicit parameters and bounds for affine-invariant and BCH codes, enhancing understanding of their structure and performance.

Contribution

It establishes a new combinatorial result on descendant set sizes, leading to explicit formulas for code dimensions and improved bounds on minimum distances for BCH codes.

Findings

Determined the size of descendant sets in cyclotomic cosets.

Derived explicit formulas for the dimensions of affine-invariant and cyclic codes.

Provided improved lower bounds on the minimum distances of dual codes.

Abstract

Affine-invariant codes have attracted considerable attention due to their rich algebraic structure and strong theoretical properties. In this paper, we study a family of affine-invariant codes whose defining set consists of all descendants of elements in the cyclotomic coset of a single specified element. Our main contributions are as follows. First, we establish a new combinatorial result that determines exactly the size of such descendant sets, which is of independent interest in the study of cyclotomic cosets. Second, using this result, we derive explicit formulas for the dimensions of the corresponding affine-invariant codes and their associated cyclic codes, and we establish lower bounds on the minimum distances of their duals. In particular, under appropriate parameter choices, these codes yield narrow-sense primitive BCH codes and their extended counterparts. For the special class of narrow-sense primitive BCH codes with designed distance $\delta = (b+1)q^{m-t-1}$, where $1 \leq b \leq q-1$ and $0 \leq t \leq m-1$, we provide exact dimension formulas and an improved lower bound on the minimum distance. The results presented here extend and sharpen several previously known results, and provide refined tools for the parametric analysis of BCH codes and their duals.