50 research outputs found
Classification of quaternary Hermitian self-dual codes of length 20
A classification of quaternary Hermitian self-dual codes of length 20 is
given. Using this classification, a classification of extremal quaternary
Hermitian self-dual codes of length 22 is also given.Comment: 9 pages. To appear in IEEE Transactions on Information Theor
Classification of generalized Hadamard matrices H(6,3) and quaternary Hermitian self-dual codes of length 18
All generalized Hadamard matrices of order 18 over a group of order 3,
H(6,3), are enumerated in two different ways: once, as class regular symmetric
(6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a
group of order 3 acting semi-regularly on points and blocks, and secondly, as
collections of full weight vectors in quaternary Hermitian self-dual codes of
length 18. The second enumeration is based on the classification of Hermitian
self-dual [18,9] codes over GF(4), completed in this paper. It is shown that up
to monomial equivalence, there are 85 generalized Hadamard matrices H(6,3), and
245 inequivalent Hermitian self-dual codes of length 18 over GF(4).Comment: 17 pages. Minor revisio
Semifields, relative difference sets, and bent functions
Recently, the interest in semifields has increased due to the discovery of
several new families and progress in the classification problem. Commutative
semifields play an important role since they are equivalent to certain planar
functions (in the case of odd characteristic) and to modified planar functions
in even characteristic. Similarly, commutative semifields are equivalent to
relative difference sets. The goal of this survey is to describe the connection
between these concepts. Moreover, we shall discuss power mappings that are
planar and consider component functions of planar mappings, which may be also
viewed as projections of relative difference sets. It turns out that the
component functions in the even characteristic case are related to negabent
functions as well as to -valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite
fields", 09-13 December 2013, Linz, Austria. This article will appear in the
proceedings volume for this workshop, published as part of the "Radon Series
on Computational and Applied Mathematics" by DeGruyte
Construction of quasi-cyclic self-dual codes
There is a one-to-one correspondence between -quasi-cyclic codes over a
finite field and linear codes over a ring . Using this correspondence, we prove that every
-quasi-cyclic self-dual code of length over a finite field
can be obtained by the {\it building-up} construction, provided
that char or , is a prime , and
is a primitive element of . We determine possible weight
enumerators of a binary -quasi-cyclic self-dual code of length
(with a prime) in terms of divisibility by . We improve the result of
[3] by constructing new binary cubic (i.e., -quasi-cyclic codes of length
) optimal self-dual codes of lengths (Type I), 54 and
66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and
60. When , we obtain a new 8-quasi-cyclic self-dual code
over and a new 6-quasi-cyclic self-dual code over
. When , we find a new 4-quasi-cyclic self-dual
code over and a new 6-quasi-cyclic self-dual code
over .Comment: 25 pages, 2 tables; Finite Fields and Their Applications, 201
Classification of the Zâ‚‚ Zâ‚„-linear Hadamard codes and their automorphism groups
Combinatorics, Coding and Security Group (CCSG)A Z₂ Z₄-linear Hadamard code of length α + 2β = 2t is a binary Hadamard code, which is the Gray map image of a Z₂ Z₄-additive code with α binary coordinates and β quaternary coordinates. It is known that there are exactly ⌊t-1/2⌋ and ⌊t/2⌋ nonequivalent Z₂ Z₄-linear Hadamard codes of length 2t, with α = 0 and α ≠0, respectively, for all t ≥ 3. In this paper, it is shown that each Z₂ Z₄-linear Hadamard code with α = 0 is equivalent to a Z₂ Z₄-linear Hadamard code with α ≠0, so there are only ⌊t/2⌋ nonequivalent Z₂ Z₄-linear Hadamard codes of length 2t. Moreover, the order of the monomial automorphism group for the Z2Z4-additive Hadamard codes and the permutation automorphism group of the corresponding Z₂ Z₄-linear Hadamard codes are given