On q-ary shortened-1-perfect-like codes
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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. Keywords: Hamming graph; multifold packings; multiple coverings; perfect codes.
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