Remarks on s-extremal codes

We study s-extremal codes over 𝔽₄ or over 𝔽₂. A Type I self-dual code over 𝔽₄ or over 𝔽₂ of length n and minimum distance d is s-extremal if the minimum weight of its shadow is largest possible. The purpose of this paper is to give some results which are missing in a series of papers by Bachoc and Gaborit [2], by Gaborit [6], and by Bautista, et. al. [1]. In particular, we give an explicit formula for the numbers of the first four nonzero weights of an s-extremal code over 𝔽₄. We improve a bound on the length for which there exists an s-extremal code over 𝔽₄ (res. 𝔽₂) with even minimum distance d (resp. d ≡ 0 (mod 4)), and give codes related to s-extremal binary codes.