On some Optimal (n,t,1,2) and (n,t,1,3) Super Imposed Codes

Abstract

One of the main problems in the theory of superimposed codes is to find the minimum length N for which an (N, T,w, r) superimposed code exists for given values of T , w and r. Let N(T,w, r) be the minimum length N for which an (N, T,w, r) superimposed code exists. The (N, T,w, r) superimposed code is called optimal when N = N(T,w, r). The values of N(T, 1, 2) are known for T ≤ 12 and the values of N(T, 1, 3) are known for T ≤ 20. In this work the values of N(T, 1, 2) for 13 ≤ T ≤ 20 and the value of N(21, 1, 3) are obtained. The optimal superimposed codes with parameters (9, 10, 1, 2), (10, 13, 1, 2), (11, 14, 1, 2), (11, 15, 1, 2), (11, 16, 1, 2) and (11, 17, 1, 2) are classified up to equivalence. The optimal (N, T, 1, 3) superimposed codes for T ≤ 20 are classified up to equivalence.