On q-ary Propelinear Perfect Codes Based on Regular Subgroups of the General Affine Group

A code is said to be propelinear if its automorphism group contains a subgroup acting on its codewords regularly. A subgroup of the group \(GA(r,q)\) of affine transformations is said to be regular if it acts regularly on vectors of \(\mathbb{F}_q^r\). Every automorphism of a regular subgroup of the general affine group \(GA(r,q)\) induces a permutation on the cosets of the Hamming code of length \(\frac{q^r-1}{q-1}\). Based on this permutation, we propose a construction of \(q\)⁠-⁠ary propelinear perfect codes of length \(\frac{q^{r+1}-1}{q-1}\). In particular, for any prime \(q\) we obtain an infinite series of almost full rank \(q\)⁠-⁠ary propelinear perfect codes.