Perfect codes and balanced generalized weighing matrices
A balanced generalized weighing matrix \(\text{BGW} (m,k,\mu)\) over a multiplicative group \(G\) is an \(m\times m\) matrix \(W=(g_{ij})\) with entries from \(G\cup\{0\}\) such that each row of \(W\) contains exactly \(k\) nonzero entries and for every \(a,b\in \{1,\dots,m\}\), \(a\neq b\), the multiset \(\{g_{ai} g_{bi}^{-1}: 1\leq i\leq m,g_{ai},g_{bi}\neq 0\}\) contains exactly \(\mu/ | G |\) copies of each element of \(G\). It is proved that any \((q^d-1)/(q-1) \times (q^d-1)/(q-1)\) matrix \(M\) with rows a set of representatives of the \((q^d -1)/(q-1)\) distinct one-dimensional codewords of the simplex code is a balanced generalized weighing matrix with parameters \[ m={q^d-1\over q-1},\;k=q^{d-1},\quad \mu=q^{d-1}-q^{d-2} \] over the multiplicative group \(GF(q)^*\) of \(GF(q)\). It is further shown that any balanced generalized weighing matrix \(M\) over \(GF(q)^*\) with these parameters has \(\text{rank}_qM\geq d\). Moreover, the equality \(\text{rank}_qM=d\) holds if and only if \(M\) is monomially equivalent to a matrix obtained by the construction above (i.e., by permutations of rows and columns and multiplying rows and columns by nonzero elements from \(GF(q))\).
scientific article; zbMATH DE number 1398789