New Linear Codes over Z p s via the Trace Map Asha Rao ∗ and Nimalsiri Pinnawala ∗ Formerly A. Baliga School of Mathematical and Geospatial Sciences RMIT University GPO Box 2476V, Melbourne, VIC - 3001, Australia Email: asha@rmit.edu.au, nimalsiri.pinnawala@rmit.edu.au Abstract— The trace map has been used very successfully to generate cocyclic complex and Butson Hadamard matrices and simplex codes over Z4 and Z2 s . We extend this technique to obtain new linear codes over Zp s . It is worth nothing here that these codes are cocyclic but not simplex codes. Further we find that the construction method also gives Butson Hadamard matrices of order p sm . I. I NTRODUCTION The use of the cocyclic map to find codes was first done in [1]. The internal structure of the Hadamard matrices used to generate these codes came from the nature of the cocyclic map, which allowed for substantial cut-downs in the computational times required to generate the Hadamard matrices and then the codes. This property of cocycles was further exploited in [2] where the authors constructed cocyclic complex and Butson Hadamard matrices via the trace map. An interesting by-product of this was the uniform construction of cocyclic codes over Z 4 and Z 2 s . These cocyclic codes were found to be simplex codes of type α. A natural extension of this research would be to find codes over Z p s . In this paper the trace map is used to define a cocycle and the cocyclic matrix obtained is found to give Butson Hadamard matrices of order p sm . In [3] Klappenecker and Roetteler use the trace map in a similar manner to obtain q + 1 mutually unbiased bases, where q is an odd prime power. The authors are not aware of the trace map being used before in this manner to find codes. A linear code C of length n over Z p s is an additive sub group of Z n p s . An element of C is called a codeword and a generator matrix of C is a matrix whose rows generate C. The Hamming weight W H (x) of an n-tuple x in Z n p s is the number of nonzero components and the Lee weight W L (x) of x =(x 1 ,x 2 ,...,x n ) is ∑ n i=1 min {x i ,p s − x i }. The Hamming and Lee distance between x, y ∈ Z n p s are defined and denoted as d H (x, y)= W H (x − y) and d L (x, y)= W L (x − y) respectively. Parameters of a linear code over Z p s are denoted by [n, k, d L ], where n is the length of the code, k is the p-dimension of the code (see [4]) and d L is the minimum Lee distance of the code. If G is a finite group (written multiplicatively with identity 1) and C is an abelian group, a cocycle (over G) is a set mapping ϕ : G × G → C which satisfies ϕ(a, b)ϕ(ab, c)= ϕ(a, bc)ϕ(b, c), ∀a, b, c ∈ G. A cocycle is normalized if ϕ(1, 1) = 1. A cocycle may be represented as a cocyclic matrix M ϕ =[ϕ(a, b)] a,b∈G once an indexing of the elements of G has been chosen. In [5], Horadam and Perera define a code over a ring R as a cocyclic code if it can be constructed by using a cocycle or the rows of a cocyclic matrix or is equivalent to such a code. Let ω = exp( 2πi k ) be the complex kth root of unity and C k = {1,ω,ω 2 ,...,ω k−1 } be the multiplicative group of all complex kth roots of unity. A square matrix H =[h i,j ] of order n with elements from C k is called a Butson Hadamard matrix if and only if HH ∗ = nI , where H ∗ is the conjugate transpose of H. A Butson Hadamard matrix is denoted by B(n, k) and in the case k =2 and k =4, B(n, k) is a Hadamard and a complex Hadamard matrix respectively. The matrix E =[e i,j ],e i,j ∈ Z k , which is obtained from H = [ω ei,j ]=[h i,j ] is called the exponent matrix associated with H. A code C over Z p , p-prime, is called a simplex code if every pair of codewords are the same Hamming distance apart. In [4] Gupta introduced the simplex code of type α and β over Z 4 and Z 2 s and in [6] Gupta et. al. constructed the senary simplex codes of type α, β and γ . A major distinguish characteristic of a simplex code of type α over either Z 4 , Z 2 s or Z 6 is that each row of its generator matrix contains every element of the alphabet equally often (see [4], [6], etc.). We construct a code over Z p s with a similar type of generator matrix, but this is not a simplex code over Z p s for p> 2 and s> 1. However in the case of s =1 this gives the usual simplex code over Z p and when p =2 and s =1, we get the binary simplex code. In Section II of this paper we outline the theory of the Galois ring GR(p s ,m) and define the trace map over GR(p s ,m). In Section III the trace map is used to define a cocycle over GR(p s ,m) and this cocycle is then used to construct a Butson Hadamard matrix H of order p sm . The rows of the exponent matrix of H form a  p sm , m, p s(m−1)  p 2s −p 2(s−1) 4  linear code over Z p s . II. GALOIS RING GR(p s , m) AND THE TRACE MAP To be able to define the cocycle, we first need to look at the definition of a Galois ring GR(p s ,m). Let p> 2 be a prime and s a positive integer. The ring of integers modulo p s is the set Z p s = {0, 1, 2,...,p s − 1}. Let h(x) ∈ Z p s [x] be a monic basic irreducible polynomial of degree m that divides (x p m −1 − 1). The Galois ring of characteristic p s and dimension m is defined to be the quotient Authorized licensed use limited to: IEEE Xplore. Downloaded on November 18, 2008 at 18:23 from IEEE Xplore. Restrictions apply.