A characterization of some {3v1 + v3, 3v0 + v2; 3, 3}-minihypers and its applications to error-correcting codes

Recently, Hamada et al. (Des. Codes Cryptogr. 2(1992), 225–229) showed that there exists an 87,5,57;3]-code meeting the Grismer bound. But it is unknown whether or not an 87,5,57;3]-code is unique up to equivalence. In order to characterize all 87,5,57;3]-codes, it is sufficient to characterize all {2v₂ + 2v₃,2v₁ + 2v₂; 4, 3}-minihypers and in order to characterize all {2v₂ + 2v₃,2v₁ + 2v₂;4,3}-minihypers, it is necessary to characterize all {3v₁ + v₃,3v₀ + v₂;3,3}-minihypers, where v₀ = 0, v₁ = 1, v₂ = 4 and v₃ = 13 (cf. Hamada and Helleseth, J. Statist. Plann. Inference, 56 (1996)). The purpose of this paper is to characterize all {3v₁ + v₃,3v₀ + v₂;3,3}-minihypers.