On covering radius of a family of codes Cm U (1 + Cm) with maximum distance between Cm and 1 + Cm

Abstract

Let C_m be the binary linear code with parity-check matrix H _m whose columns are all distinct nonzero weight-2 length-m binary vectors. The all-one binary vector 1 of length (^m ₂) has maximum distance from C^⊥ _mm, the dual of C _m, that is the covering radius p(C^⊥ _m) of C^⊥ _m is d(l, C^⊥ _m). The covering radius of C̄^⊥ _m := C ^⊥ _m U (1 + C^⊥ _m) and C̄_m, the dual of C̄^⊥ _m is also considered. It is shown that p(C̄_m) = p(C_m) + 1 = [m+/2] = p(C_m+2) for m ≥ 4. It is conjectured that p(C̄^⊥ _m) = P(C^⊥ _m-1) for m ≥7. This equality holds for 7 ≤ m ≤ 10, and the codes C̄^⊥ _m, 4 ≤m ≤ 10, are good covering codes.