Completely regular codes with different parameters and the same distance-regular coset graphs

TL;DR

This paper constructs various classes of completely regular codes with identical intersection arrays over different fields, explores their connection to distance-regular graphs, and introduces new families of uniformly packed codes.

Contribution

It presents a novel Kronecker construction for completely regular codes over different alphabets and links these codes to distance-regular graphs, expanding the understanding of their parameters.

Findings

Constructed completely regular codes over different fields with identical intersection arrays.

Connected these codes to distance-regular bilinear forms graphs as coset graphs.

Provided explicit constructions of infinite families of uniformly packed codes that are not completely regular.

Abstract

A known Kronecker construction of completely regular codes has been investigated taking different alphabets in the component codes. This approach is also connected with lifting constructions of completely regular codes. We obtain several classes of completely regular codes with different parameters, but identical intersection array. Given a prime power $q$ and any two natural numbers $a,b$, we construct completely transitive codes over different fields with covering radius $\rho=\min\{a,b\}$ and identical intersection array, specifically, one code over $\F_{q^r}$ for each divisor $r$ of $a$ or $b$. As a corollary, for any prime power $q$, we show that distance regular bilinear forms graphs can be obtained as coset graphs from several completely regular codes with different parameters. Under the same conditions, an explicit construction of an infinite family of $q$-ary uniformly packed codes (in the wide sense) with covering radius $\rho$, which are not completely regular, is also given.