On \((p,1)\)-subnormal \(q\)-ary covering codes (Q1331896)

A \(q\)-ary code of length \(n\) with \(M\) codewords and covering radius \(R\) is called a \((q,n,M)R\) code. A \((q,n,M)R\) code \(C\) has \((k,t)\)-subnorm \(S\) if there is a partition of \(C\) into \(k\) nonempty subsets \(C_ 1,\dots, C_ k\) such that \(\min_ a d(x,C_ a)+ \max_ a d(x,C_ a)\leq S\) whenever \(R-t\leq d(x,C) \leq R\). If \(C\) has \((k,t)\)-subnorm \(2R+1\) it is called \((k,t)\)-subnormal. An \([n,k]R\) code is a linear code with length \(n\), dimension \(k\) and covering radius \(R\). The author shows that if \(q\neq 3\) is a prime power and there exists a \((q,n,M)1\) code then there exists also a \((q,1)\)-subnormal \((q,qn+1, q^{(q-1)n} M)1\) code. The author also shows that all nontrivial linear \(q\)-ary codes with covering radius 1 are \((q,1)\)-subnormal with the exception of the ternary \([4,2]1\) Hamming code.