A note on the Assmus--Mattson theorem for some binary codes II

TL;DR

This paper extends the analysis of four-weight binary codes supporting t-designs, demonstrating similar bounds for five and six-weight codes, thereby advancing understanding of their combinatorial structure.

Contribution

It generalizes previous results by establishing bounds on t for five and six-weight binary codes supporting t-designs.

Findings

t ≤ 5 for four-weight codes

Analogous bounds established for five and six-weight codes

Supports the design theory of binary codes

Abstract

Let $C$ be a four-weight binary code, which has all one vector. Furthermore, we assume that $C$ supports $t$-designs for all weights obtained from the Assmus--Mattson theorem. We previously showed that $t\leq 5$. In the present paper, we show an analogue of this result in the cases of five and six-weight codes.