Few-weight codes from trace codes over a local ring

Abstract

In this paper, new few weights linear codes over the local ring R = F p +uF p +vF p +uvF p , with u 2 = v 2 = 0, uv = vu, are constructed by using the trace function defined over an extension ring of degree m. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gaussian sums over finite fields. Two different defining sets are explored. Using a linear Gray map from R to F 4 p , we obtain several families of new p-ary codes from trace codes of dimension 4m. For the first defining set: when m is even, or m is odd and p ≡ 3 (mod 4), we obtain a new family of two-weight codes, which are shown to be optimal by the application of the Griesmer bound; when m is even and under some special conditions, we obtain two new classes of three-weight codes. For the second defining set: we obtain a new class of two-weight codes and prove that it meets the Griesmer bound. In addition, we give the minimum distance of the dual code. Finally, applications of the p-ary image codes in secret sharing schemes are presented.

Optimal Few-Weight Codes From Simplicial Complexes

“…(1) If |A| = 1, |B| = 2 and A ∩ B = ∅, then φ(C L 2 ) is a four-weight binary code with parameters [48, 6,20] and weight enumerator 1 + 6z 20 + 54z 24 + z 32 + 2z 36 .…”

Cyclotomic Trace Codes

“…[23]). In fact, the code [208, 4, 160] 5 is four copies of the above mentioned projective [52, 4,40] code. SRG(14641,4880,1599,1640) is isomorphic to the graph obtained by CY2 construction (see Reference [23]) for q = 11 and S = 4.…”

“…The SRG with parameters (81,16,7,2) is isomorphic to the SRG obtained from the projective ternary [8,4] code with weights 3, 6 (see Reference [13]). The code [16,4,6] 3 is two copies of the above-mentioned projective [8,4,3] code. Further, we have constructed an SRG with parameters (361,72,23,12), and according to Brouwer's table (see Reference [22]), a known graph with these parameters is obtainable from the orthogonal array.…”

“…In Reference [1], Ding introduced a generic construction of linear codes using a defining set D and the trace function. Many classes of known codes can be produced by choosing an appropriate defining set [2][3][4][5][6]. Some of these codes support combinatorial 2-designs [7,8].…”

“…In particular, we obtain projective codes over GF(4) that give rise to optimal or near optimal quantum codes over GF (4).…”