Further results on the covering radius of small codes

Abstract

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 ) . In the paper, necessary and sufficient conditions for K ( t , b , R ) = M are given for M = 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 ⩽ 5 were settled in an earlier study by the same authors. For binary codes, it is proved that K ( 0 , 2 b + 4 , b ) ⩾ 9 for b ⩾ 1 . For ternary codes, it is shown that K ( 3 t + 2 , 0 , 2 t ) = 9 for t ⩾ 2 . New upper bounds obtained include K ( 3 t + 4 , 0 , 2 t ) ⩽ 36 for t ⩾ 2 . Thus, we have K ( 13 , 0 , 6 ) ⩽ 36 (instead of 45, the previous best known upper bound).