On perfect codes: Rank and kernel (Q1866012)

The rank of a nonlinear binary code \(C\) is the dimension of the subspace spanned by \(C\). The kernel of \(C\) is the largest possible linear code \(C'\) such that \(C\) can be obtained as a union of cosets of \(C'\). The authors study the problem of determining for what parameters \((r,k)\) there exists a perfect binary one-error-correcting code of length \(n = 2^m-1\), rank \(r\), and kernel dimension \(k\), and obtain several bounds on these parameters.