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We study codes with parameters of $q$ -ary shortened Hamming codes, i.e., $(n=(q^{m}-q)/(q-1), q^{n-m}, 3)_{q}$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. 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. Secondly, for every admissible length starting from $n=20$ , 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. [ABSTRACT FROM AUTHOR]

Diameter perfect codes form a natural generalization for perfect codes. They are based on the code-anticode bound which generalizes the sphere-packing bound. The code-anticode bound was proved by Delsarte for distance-regular graphs and it holds for some other metrics too. In this paper we prove the bound for non-binary constant-weight codes with the Hamming metric and characterize the diameter perfect codes and the maximum size anticodes for these codes. We distinguish between six families of non-binary diameter constant-weight codes and four families of maximum size non-binary constant-weight anticodes. Each one of these families of diameter perfect codes raises some different questions. We consider some of these questions and leave lot of ground for further research. Finally, as a consequence, some $t$ -intersecting families related to the well-known Erdös-Ko-Rado theorem, are constructed. [ABSTRACT FROM AUTHOR]

An additive quaternary [n,k,d]-code (length n, quaternary dimension k, minimum distance d) is a 2k-dimensional F2-vector space of n-tuples with entries in F2 ⊕ F2 (the 2-dimensional vector space over F2) with minimum Hamming distance d. We determine the optimal parameters of additive quaternary codes of dimension k ≤ 3. The most challenging case is dimension k = 2.5. We prove that an additive quaternary [n,2.5,d]-code where d < n−1 exists if and only if 3(n−d) ≥ ⌈d/2⌉ + ⌈d/4⌉ + ⌈d/8⌉. In particular, we construct new optimal 2.5-dimensional additive quaternary codes. As a by-product, we give a direct proof for the fact that a binary linear [3m,5,2e]2-code for e < m−1 exists if and only if the Griesmer bound 3(m−e) ≥e/2⌉ + ⌈e/4⌉ + ⌈e/8⌉ is satisfied. [ABSTRACT FROM AUTHOR]