ELEMENTARY p-GROUPS WITH THE RÉDEI PROPERTY

Abstract

It is known that an elementary 3-group of rank k has the Rédei property for k ≤ 4 and does not have the Rédei property for k ≥ 6. The last open case, rank 5, is settled here by using a computer-aided approach to show that such a group has the Rédei property.

Projective tilings and full-rank perfect codes

“…The examples of codes are available at https://dx.doi.org/10.21227/w856-4b70. As follows from [8], 7 is the maximum value of kernel dimension for full-rank 1-perfect codes in F 13 3 , and so only the existence of codes of kernel dimension less than 3 remains unsolved for these parameters (both for full-rank codes and for codes of rank 12, see the rank-kernel table in [12]).…”

SMALL p-GROUPS WITH FULL-RANK FACTORIZATION

“…Form the family of the subsets Ag, g ∈ G, and search for members of this family that form a partition of G. The general problem of finding a partition of a set given a family of subsets of the set is known as the exact cover problem, so we here have instances of that problem and can apply known algorithms. In [8] and [7] the exact cover approach was used to study the Rédei property of elementary 2-groups and elementary 3-groups, respectively.…”

An enumeration of 1-perfect ternary codes

“…In the binary case, most of related questions have been solved [4], [7], [32], [17], including the characterization of all admissible rank-kernel dimension pairs for 1-perfect codes [3]. For q = 3 (we mention important results [16,30] focused on this case), many questions are open, and the table of ranks and kernels for concatenated codes (Table 1) can be considered as a step in this direction.…”