The number of self-dual codes over $${Z_{p^3}}$$

Abstract

Let p be a prime number. In this paper, we consider codes over the ring Z p 3 of integers modulo p 3 and give a characterization of self-duality. This leads to a construction of self-dual codes and a mass formula, which counts the number of such codes over Z p 3 .

Mass Formula for Self-Dual Codes over Galois Rings GR(p3, r)

“…Therefore, we have p rk( n 2 −1) possible choices for the matrices A 31 , A 41 and A 42 . We have proved the following result, which is analogous to Proposition 2.2 of [10]. Proposition 2.…”

“…For the remainder of this paper, we assume that p is an odd prime. Following the argument in Section 2.1 of [10], there are p rkl possible choices for A 31 , p rk(k−1) 2 for A 41 and p rk(k−1) 2 for A 42 . Therefore, we have p rk( n 2 −1) possible choices for the matrices A 31 , A 41 and A 42 .…”

“…Mass formula for self-dual codes over the ring Z p e for any prime p and for any positive integer e are established by the effort of many authors [1,5,[9][10][11]. A classification method of self-dual codes over Z m for arbitrary integer m is given in [13].…”

MASS FORMULA OF SELF-DUAL CODES OVER GALOIS RINGS GR(p2, 2)

“…[8,11,14,15] Let σ q (n, k) be the number of self-orthogonal codes of length n and dimension k over F q , where q = p m for some prime p and an integer m. Then:…”

“…In 1996, Gaborit calculated the mass formulas for self-dual codes over Z 4 in [4]. This paper motivated Nagata, et al to find the mass formulas for self-dual codes over Z p e in consecutive papers, [1], [10], [11], [12].…”