29,187 research outputs found
Additive Asymmetric Quantum Codes
We present a general construction of asymmetric quantum codes based on
additive codes under the trace Hermitian inner product. Various families of
additive codes over \F_{4} are used in the construction of many asymmetric
quantum codes over \F_{4}.Comment: Accepted for publication March 2, 2011, IEEE Transactions on
Information Theory, to appea
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
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
The Perfect Binary One-Error-Correcting Codes of Length 15: Part II--Properties
A complete classification of the perfect binary one-error-correcting codes of
length 15 as well as their extensions of length 16 was recently carried out in
[P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary
one-error-correcting codes of length 15: Part I--Classification," IEEE Trans.
Inform. Theory vol. 55, pp. 4657--4660, 2009]. In the current accompanying
work, the classified codes are studied in great detail, and their main
properties are tabulated. The results include the fact that 33 of the 80
Steiner triple systems of order 15 occur in such codes. Further understanding
is gained on full-rank codes via switching, as it turns out that all but two
full-rank codes can be obtained through a series of such transformations from
the Hamming code. Other topics studied include (non)systematic codes, embedded
one-error-correcting codes, and defining sets of codes. A classification of
certain mixed perfect codes is also obtained.Comment: v2: fixed two errors (extension of nonsystematic codes, table of
coordinates fixed by symmetries of codes), added and extended many other
result
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
New extremal binary self-dual codes of length 68 via short kharaghani array over f_2 + uf_2
In this work, new construction methods for self-dual codes are given. The
methods use the short Kharaghani array and a variation of it. These are
applicable to any commutative Frobenius ring. We apply the constructions over
the ring F_2 + uF_2 and self-dual Type I [64, 32, 12]_2-codes with various
weight enumerators obtained as Gray images. By the use of an extension theorem
for self-dual codes we were able to construct 27 new extremal binary self-dual
codes of length 68. The existence of the extremal binary self-dual codes with
these weight enumerators was previously unknown.Comment: 10 pages, 5 table
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