The nonexistence of some optimal ternary codes of dimension five (Q1281411)

The author considers the problem of finding the smallest possible length, denoted by \(n_q(k,d)\), of a \(q\)-ary linear code of dimension \(k\) and minimum distance \(d\). A well-known bound on \(n_q(k,d)\) is the Griesner bound: \[ n_q(k,d)\geq g_q(k,d)= \sum^{k-1}_{i=0} \left\lceil {d\over q^i} \right \rceil. \] Given a prime power \(q\) and a positive integer \(k\), there exist \(q\)-ary, \(k\)-dimensional codes, meeting this bound, for all but a finite number of \(d\)'s. Here the author proves the nonexistence of codes with parameters \([143, 5, 94]_3\) and \([147,5,97]_3\). This determines the exact value of \(n_3(5,d)\) for \(d=94\), 95, 96, 97, 98, 99. These were the only \(d\)'s for which the exact value of \(n_3(5,d)\) was not known. It has to be noted that the nonexistence of \([148, 5,98]_3\) and \([149,5,99]_3\) codes follows also from a result proved independently by M. van Eupen and by R. Hill and P. Lizak.