553 research outputs found
The MacWilliams Identity for the Skew Rank Metric
The weight distribution of an error correcting code is a crucial statistic in
determining it's performance. One key tool for relating the weight of a code to
that of it's dual is the MacWilliams Identity, first developed for the Hamming
metric. This identity has two forms: one is a functional transformation of the
weight enumerators, while the other is a direct relation of the weight
distributions via (generalised) Krawtchouk polynomials. The functional
transformation form can in particular be used to derive important moment
identities for the weight distribution of codes. In this paper, we focus on
codes in the skew rank metric. In these codes, the codewords are skew-symmetric
matrices, and the distance between two matrices is the skew rank metric, which
is half the rank of their difference. This paper develops a -analog
MacWilliams Identity in the form of a functional transformation for codes based
on skew-symmetric matrices under their associated skew rank metric. The method
introduces a skew- algebra and uses generalised Krawtchouk polynomials.
Based on this new MacWilliams Identity, we then derive several moments of the
skew rank distribution for these codes.Comment: 39 page
The MacWilliams Identity for the Hermitian Rank Metric
Error-correcting codes have an important role in data storage and
transmission and in cryptography, particularly in the post-quantum era.
Hermitian matrices over finite fields and equipped with the rank metric have
the potential to offer enhanced security with greater efficiency in encryption
and decryption. One crucial tool for evaluating the error-correcting
capabilities of a code is its weight distribution and the MacWilliams Theorem
has long been used to identify this structure of new codes from their known
duals. Earlier papers have developed the MacWilliams Theorem for certain
classes of matrices in the form of a functional transformation, developed using
-algebra, character theory and Generalised Krawtchouk polynomials, which is
easy to apply and also allows for moments of the weight distribution to be
found. In this paper, recent work by Kai-Uwe Schmidt on the properties of codes
based on Hermitian matrices such as bounds on their size and the eigenvalues of
their association scheme is extended by introducing a negative- algebra to
establish a MacWilliams Theorem in this form together with some of its
associated moments. The similarities in this approach and in the paper for the
Skew-Rank metric by Friedlander et al. have been emphasised to facilitate
future generalisation to any translation scheme.Comment: 39 pages. arXiv admin note: substantial text overlap with
arXiv:2210.1615
Partitions of Matrix Spaces With an Application to -Rook Polynomials
We study the row-space partition and the pivot partition on the matrix space
. We show that both these partitions are reflexive
and that the row-space partition is self-dual. Moreover, using various
combinatorial methods, we explicitly compute the Krawtchouk coefficients
associated with these partitions. This establishes MacWilliams-type identities
for the row-space and pivot enumerators of linear rank-metric codes. We then
generalize the Singleton-like bound for rank-metric codes, and introduce two
new concepts of code extremality. Both of them generalize the notion of MRD
codes and are preserved by trace-duality. Moreover, codes that are extremal
according to either notion satisfy strong rigidity properties analogous to
those of MRD codes. As an application of our results to combinatorics, we give
closed formulas for the -rook polynomials associated with Ferrers diagram
boards. Moreover, we exploit connections between matrices over finite fields
and rook placements to prove that the number of matrices of rank over
supported on a Ferrers diagram is a polynomial in , whose
degree is strictly increasing in . Finally, we investigate the natural
analogues of the MacWilliams Extension Theorem for the rank, the row-space, and
the pivot partitions
Classification of poset-block spaces admitting MacWilliams-type identity
In this work we prove that a poset-block space admits a MacWilliams-type
identity if and only if the poset is hierarchical and at any level of the
poset, all the blocks have the same dimension. When the poset-block admits the
MacWilliams-type identity we explicit the relation between the weight
enumerators of a code and its dual.Comment: 8 pages, 1 figure. Submitted to IEEE Transactions on Information
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