Enumeration of some matrices and free linear codes over commutative finite local rings (Q2062667)

This work generalizes results of \textit{K. Abdel-Ghaffar} [Linear Algebra Appl. 436, No. 7, 2665--2669 (2012; Zbl 1236.15043)], who enumerated \(m\times n\) matrices of rank \(r\) over finite fields having a given number of rows of single unit. A vector of single unit is a vector with exactly one entry equal to a unit and all other entires equal to \(0\). In this article, the setting is generalized to finite commutative local rings \(R\). Such rings have the advantage that one can work in the residue field and lift the enumeration results back to \(R\) with some modifications. The author enumerates the number of \(m\times n\) matrices of McCoy rank \(r\) (see [\textit{N. H. McCoy} [Rings and ideals. La Salle, IL: Open Court Publishing (1948; Zbl 0041.36406)]) over \(R\) with exactly \(k\) rows of unit weight and, furthermore, counts the number of free linear codes of \(R^n\) of rank \(r\) containing exactly \(k\) standard basis vectors. Free linear codes of \(R^n\) are free submodules of \(R^n\). The formulae for these two enumeration problems closely resemble their finite field counterparts.