On $q$-ary shortened-$1$-perfect-like codes

Abstract

We study codes with parameters of q-ary shortened Hamming codes, i.e., (n = (q m − q)/(q − 1), q n−m , 3) q . At first, we prove the fact mentioned in [A. E. Brouwer et al. Bounds on mixed binary/ternary codes. IEEE Trans. Inf. Theory 44 (1998) 140-161] that such codes are optimal, generalizing it to a bound for multifold packings of radius-1 balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal q-fold packing with minimum distance 2. At second, we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code.