On perfect p-ary codes of length p+1
Let p be a prime number and assume p >= 5. We will use a result of L. Redei to prove, that every perfect 1-error correcting code C of length p + 1 over an alphabet of cardinality p, such that C has a rank equal to p and a kernel of dimension p - 2, will be equivalent to some Hamming code H. Further, C can be obtained from H, by the permutation of the symbols, in just one coordinate position.
perfect codes redei theorem