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s-extremal additive $\mathbb F_4$ codes

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  • Binary self-dual codes and additive self-dual codes over $\mathbb F_4$ have in common interesting properties, for example, Type I, Type II, shadows, etc. Recently Bachoc and Gaborit introduced the notion of $s$-extremality for binary self-dual codes, generalizing Elkies' study on the highest possible minimum weight of the shadows of binary self-dual codes. In this paper, we introduce a concept of $s$-extremality for additive self-dual codes over $\mathbb F_4$, give a bound on the length of these codes with even distance $d$, classify them up to minimum distance $d = 4$, give possible lengths and (shadow) weight enumerators for which there exist $s$-extremal codes with $5 \leq d \leq 11$ and give five $s$-extremal codes with $d = 7$. We construct four $s$-extremal codes of length $n = 13$ and minimum distance $d = 5$. We relate an $s$-extremal code of length $3d$ to another $s$-extremal code of that length, and produce extremal Type II codes from $s$-extremal codes.
    Mathematics Subject Classification: Primary: 94B05, 94B65; Secondary: 05B05, 11T71.

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