Aleksandr Aleksandrovich Nechaev

Complete weight enumerators of generalized Kerdock code and related linear codes over Galois ring

Abstract A generalized Kerdock code is a nonlinear (n,n 2 ,[(q−1)/q](n− n )) -code of length n=qm+1 over the field of q=2l elements (l⩾1,m is odd). It is a concatenation of some special (base) linear code over the Galois ring of characteristic 4 and Reed–Solomon code of dimension 2. Here the complete weight enumerators of Kerdock code, base linear code and their analogues for even m are described. Incidentally, the weight characteristics of linear recurrences with the distinguished characteristic polynomial over the pointed Galois ring are indicated. Methods of proofs are based on the properties of trace function in Galois ring and quadrics over the field of characteristic 2.

On Cyclic Top-Associative Generalized Galois Rings

A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic p n , for a prime number p, such that its top-factor \(\overline{S} = S/pS\) is a finite semifield. It is well known that if S is an associative Galois Ring (GR) then the set \(S^* = S \ pS\) is a finite multiplicative abelian group. This group is cyclic if and only if S is either a finite field, or a residual integer ring of odd characteristic or the ring ℤ4. A GGR is called top-associative if \(\overline{S}\) is a finite field. In this paper we study the conditions for a top-associative not associative GGR S to be cyclic.