Some results on type IV codes over Z4

Abstract

Abstract-Dougherty, Gaborit, Harada, Munemasa, and Solé have previously given an upper bound on the minimum Lee weight of a Type IV self-dual -code, using a similar bound for the minimum distance of binary doubly even self-dual codes. We improve their bound, finding that the minimum Lee weight of a Type IV self-dual -code of length is at most 4 12 , except when = 4, and = 8 when the bound is 4, and = 16 when the bound is 8. We prove that the extremal binary doubly even self-dual codes of length 24= 32 are not -linear. We classify Type IV-I codes of length 16. We prove that all Type IV codes of length 24 have minimum Lee weight 4 and minimum Hamming weight 2, and the Euclidean-optimal Type IV-I codes of this length have minimum Euclidean weight 8.

A generalized Gleason–Pierce–Ward theorem

“…Wood shows that the MacWilliams identities are satisfied for codes over finite Frobenius rings [25]. From this point of view, Type IV self-dual codes over various rings are studied by many researchers [1][2][3][4]7,[9][10][11].…”

“…The Roman numeral appropriate to C in Theorem 1.2 is customarily called the type of C. In light of the Gleason-Pierce-Ward theorem, self-dual codes over G F (2), G F (3), and G F (4) with nontrivial divisors are studied by various researchers such as, but not restricted to, [8,11,[13][14][15].…”