New code upper bounds for the folded n-cube

TL;DR

This paper develops new upper bounds for the maximum size of codes in the folded n-cube using algebraic and semidefinite programming techniques, extending previous methods applied to hypercubes.

Contribution

It introduces an extended approach to bound code sizes in folded n-cubes by block-diagonalizing the Terwilliger algebra and applying semidefinite programming.

Findings

Derived new upper bounds for $A(oxed{n},d)$

Extended Schrijver's method to folded n-cubes

Demonstrated effectiveness of algebraic techniques in coding theory

Abstract

Let $\Gamma$ denote a distance-regular graph. The maximum size of codewords with minimum distance at least $d$ is denoted by $A(\Gamma,d)$. Let $\square_n$ denote the folded $n$-cube $H(n,2)$. We give an upper bound on $A(\square_n,d)$ based on block-diagonalizing the Terwilliger algebra of $\square_n$ and on semidefinite programming.The technique of this paper is an extension of the approach taken by A. Schrijver \cite{s} on the study of $A(H(n,2),d)$.