On components of a Kerdock code and the dual of the BCH code $C_{1,3}$

TL;DR

This paper analyzes the structure of Kerdock codes and dual BCH codes, revealing their component composition and association scheme properties, with implications for coding theory and combinatorial designs.

Contribution

It provides new insights into the component structure of Kerdock and dual BCH codes, including alternative proofs of their association scheme properties.

Findings

A punctured Kerdock code has two i-components.

The dual of BCH code is a single i-component.

The restriction of the Hamming scheme to a doubly shortened Kerdock code forms an association scheme.

Abstract

In the paper we investigate the structure of $i$-components of two classes of codes: Kerdock codes and the duals of the primitive cyclic BCH code with designed distance 5 of length $n=2^m-1$, for odd $m$. We prove that for any admissible length a punctured Kerdock code consists of two $i$-components and the dual of BCH code is a $i$-component for any $i$. We give an alternative proof for the fact presented by De Caen and van Dam in 1999 that the restriction of the Hamming scheme to a doubly shortened Kerdock code is an association scheme.