Two New Families of Two-Weight Codes
Minjia Shi, Yue Guan, Patrick Solé · IEEE Transactions on Information Theory · 2017
We construct two new infinite families of trace codes of dimension 2m, over the ring Fp+ uFp, with u2= u, when p is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear p-ary codes of respective lengths (pm-1)2and 2(pm-1)2. When m is singly even, the first family gives five-weight codes. When m is odd and p ≡ 3 (mod 4), the first family yields p-ary two-weight codes, which are shown to be optimal by application of the Griesmer bound. The second family consists of two-weight codes that are shown to be optimal, by the Griesmer bound, whenever p = 3 and m ≥ 3, or p ≥ 5 and m ≥ 4. Applications to secret sharing schemes are given.