Further results on the covering radius of small codes |
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| Authors: | Gerzson Kéri Patric RJ ÖstergÅrd |
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| Institution: | a Computer and Automation Research Institute, Hungarian Academy of Sciences, H-1111 Budapest Kende u. 13-17, Hungary
b Department of Electrical and Communications Engineering, Helsinki University of Technology, P.O. Box 3000, 02015 HUT, Finland |
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| 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(0,2b+4,b)?9 for b?1. For ternary codes, it is shown that K(3t+2,0,2t)=9 for t?2. New upper bounds obtained include K(3t+4,0,2t)?36 for t?2. Thus, we have K(13,0,6)?36 (instead of 45, the previous best known upper bound). |
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| Keywords: | Covering code Covering radius Mixed code Surjective code |
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