Abstract
Twisted permutation codes, introduced recently by the second and third authors, belong to the family of frequency permutation arrays. Like some other codes in this family, such as the repetition permutation codes, they are obtained by a repetition construction applied to a smaller code (but with a "twist" allowed). The minimum distance of a twisted permutation code is known to be at least the minimum distance of a corresponding repetition permutation code, but in some instances can be larger. We construct two new infinite families of twisted permutation codes with minimum distances strictly greater than those for the corresponding repetition permutation codes. These constructions are based on two infinite families of finite groups and their representations. The first is a family of $p$-groups, for an odd prime $p$, while the second family consists of the $4$-dimensional symplectic groups over a finite field of even order. In the latter construction, properties of the graph automorphism of these symplectic groups play an important role.
Highlights
When considering the delivery of services to customers, the “last mile” refers to the often problematic last leg of the journey
It has been proposed that this “last mile” could involve communication via powerlines as an effective means of delivering reliable telecommunications at low cost [11, 15]. Such a solution requires new kinds of encoding techniques for robust communication of information, and for this purpose constant composition codes, in particular the subclass of frequency permutation arrays, have been suggested as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication [4, 5]
Given a set of n cells with distinct charge levels, the rank of a cell indicates the relative position of its own charge level, and so the ranks of the n cells induce a permutation of {1, 2, . . . , n}
Summary
When considering the delivery of services to customers, the “last mile” refers to the often problematic last leg of the journey. It has been proposed that this “last mile” could involve communication via powerlines as an effective means of delivering reliable telecommunications at low cost [11, 15] Such a solution requires new kinds of encoding techniques for robust communication of information, and for this purpose constant composition codes, in particular the subclass of frequency permutation arrays, have been suggested as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication [4, 5]. The codes we study are called twisted permutation codes They are FPAs with potentially good error-correcting properties (see [10]). In [10], the second and third authors, with Spiga, proved that the error-correcting capability of a twisted permutation code is at least as good as that of a corresponding repetition permutation code for the group, and gave examples for which it was better (see [10, Table 1]).
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