Further results on the covering radius of small codes

The minimum number of codewords in a code with t ternary and b binary coordinates and covering radius R   is denoted by K(t,b,R)K(t,b,R). In the paper, necessary and sufficient conditions for K(t,b,R)=MK(t,b,R)=M are given for M=6M=6 and 7 by proving that there exist exactly three families of optimal codes with six codewords and two families of optimal codes with seven codewords. The cases M⩽5M⩽5 were settled in an earlier study by the same authors. For binary codes, it is proved that K(0,2b+4,b)⩾9K(0,2b+4,b)⩾9 for b⩾1b⩾1. For ternary codes, it is shown that K(3t+2,0,2t)=9K(3t+2,0,2t)=9 for t⩾2t⩾2. New upper bounds obtained include K(3t+4,0,2t)⩽36K(3t+4,0,2t)⩽36 for t⩾2t⩾2. Thus, we have K(13,0,6)⩽36K(13,0,6)⩽36 (instead of 45, the previous best known upper bound).