104,399 research outputs found
Classification of extremal and -extremal binary self-dual codes of length 38
In this paper we classify all extremal and -extremal binary self-dual
codes of length 38. There are exactly 2744 extremal self-dual
codes, two -extremal codes, and 1730 -extremal
codes. We obtain our results from the use of a recursive algorithm used in the
recent classification of all extremal self-dual codes of length 36, and from a
generalization of this recursive algorithm for the shadow. The classification
of -extremal codes permits to achieve the classification of all
-extremal codes with d=6.Comment: revised version - paper submitted (4/4/2011) to IEEE trans.
Information and accepted 20/10/201
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
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 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
On the Residue Codes of Extremal Type II Z4-Codes of Lengths 32 and 40
In this paper, we determine the dimensions of the residue codes of extremal
Type II Z4-codes for lengths 32 and 40. We demonstrate that every binary doubly
even self-dual code of length 32 can be realized as the residue code of some
extremal Type II Z4-code. It is also shown that there is a unique extremal Type
II Z4-code of length 32 whose residue code has the smallest dimension 6 up to
equivalence. As a consequence, many new extremal Type II Z4-codes of lengths 32
and 40 are constructed.Comment: 19 page
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