A note on the Assmus--Mattson theorem for some ternary codes (a resume)

TL;DR

This paper analyzes when weight-based designs derived from a ternary code under the Assmus–Mattson framework are $t$-designs, focusing on two-weight and three-weight codes under the AM-condition. It establishes sharp bounds on the dual distance $d^{\perp}$ and the design parameter $t$, showing that for a two-weight code either $d^{\perp}=5$ with the dual a ternary Golay code and $t=4$, or $d^{\perp}\le 4$ with $t\le 3$, while for a three-weight code one has $d^{\perp}\le 6$ and $t\le 5$, with analogous Golay-characterizations when a weight-$n$ vector is present. A strengthened AM-result (Theorem main3) provides weight-enumerator criteria yielding $(t+1)$-designs in duals for cases $d^{\perp}-t=1,2,3$, although explicit instances are scarce. The paper derives a corollary giving a new characterization of (extended) ternary Golay codes, discusses conjectures and open questions, and reports computational checks using Magma and Mathematica, including a five-weight example with delta$$(C^{\perp})<s(C^{\perp})$$.

Abstract

Let $C$ be a two and three-weight ternary code. Furthermore, we assume that $C_\ell$ are $t$-designs for all $\ell$ by the Assmus--Mattson theorem. We show that $t \leq 5$. As a corollary, we provide a new characterization of the (extended) ternary Golay code.

Theorems & Definitions (7)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Remark 2.1
• Example 2.3