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We consider generalized Preparata codes with a noncommutative group operation. These codes are shown to induce new partitions of Hamming codes into cosets of these Preparata codes. The constructed partitions induce 2-resolvable Steiner quadruple systems S( n, 4, 3) (i.e., systems S( n, 4, 3) that can be partitioned into disjoint Steiner systems S( n, 4, 2)). The obtained partitions of systems S( n, 4, 3) into systems S( n, 4, 2) are not equivalent to such partitions previously known. [ABSTRACT FROM AUTHOR]

A method for constructing optimal almost equisymbol codes based on difference matrices is proposed. Some of these codes generate previously unknown optimal equidistant codes satisfying the Plotkin bound. As a consequence, they generate new resolvable balanced incomplete block designs. [ABSTRACT FROM AUTHOR]

The structure of all different Steiner triple systems S(2−1, 3, 2) of rank 2− m+2 over $\mathbb{F}_2 $ is described. This induces a natural recurrent method for constructing Steiner triple systems of any rank. In particular, the method gives all different such systems of order 2 − 1 and rank ≤ 2 − m + 2. The number of such different systems of order 2 − 1 and rank less than or equal to 2 − m + 2 which are orthogonal to a given code is found. It is shown that all different triple Steiner systems of order 2 − 1 and rank ≤ 2 − m + 2 are derivative and Hamming. Furthermore, all such triples are embedded in quadruple systems of the same rank and in perfect binary nonlinear codes of the same rank. [ABSTRACT FROM AUTHOR]

Inaccuracies in computations in the paper of the authors on classification of perfect binary codes of lengths 15 and 16 and of rank 13 are fixed. An explicit construction of all extended perfect codes of length 16 and rank 13 with a given kernel size is presented. Perfect binary codes of length 15 and rank 14 obtained by the general doubling construction are classified. [ABSTRACT FROM AUTHOR]

The acknowledgment (footnote to the title of the paper) should read as follows: The research was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50- 00150. [ABSTRACT FROM AUTHOR]