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.

如果一个代码的自同构群包含一个有规律地作用于其代码字的子群，则称该代码是线性的。如果仿射变换组\(GA(r,q)\)的子群有规律地作用于\(\mathbb{F}_q^r\)的向量，则称它是有规律的。一般仿射群\(GA(r,q)\)的正则子群的每一个自同构都会在长度为\(\frac{q^r-1}{q-1}\的汉明码的陪集上产生一个置换) . 基于这种排列，我们提出了长度为\(\frac{q^{r+1}-1}{q-1}\)的\(q\) ⁠-⁠ary 线性完美码的构造。特别是，对于任何素数\(q\)，我们得到一个几乎满秩的无限级数\(q\)⁠-⁠ary 线性完美代码。