Self-Dual Codes
Self-Dual Codes
A self-dual code C is a subset of R^n where R is any ring such that
C=C* where C* is the orthogonal under some inner product. Initially,
people were interested in binary sefl-dual codes. They had applications
to design theory (they were used in the proof of the non-existence of
the projective plane of order 10) and lattice theory (they can be used
to build unimodular lattices).
For an interesting conjecture see my page about the
length 72 problem.
Later interest grew in ternary and
quatenary papers. More recently self-dual codes over Z_{2k} and other
rings has grown.
You can read some of my papers on self-dual codes:
-
Shadow Codes and their Weight Enumerators -- IEEE Transactions
on Information Theory, Volume 41, Number 3, May 1995.
- Extremal Binary Self-dual Codes
-- IEEE Transactions on Information
Theory, 2036-2047, November 1997, with M. Harada and T.A. Gulliver.
- Type II Codes Over F_2 + u F_2,
IEEE-IT, vol 45. No. 1, January 1999,
with M. Harada, P. Gaborit, and P. Sole.
-
Optimal Ternary Formally Self-dual Codes, Discrete Mathematics 196, 1999,
117-135,
with T.A. Gulliver an M. Harada.
-
Type II Codes, Even Unimodular Lattices, and Invariant Rings,
IEEE-IT, Volume 45, Number 4, 1999, 1194-1205,
with E. Bannai, M. Harada, and M. Oura.
- Self-Dual Codes over Rings and the Chinese Remainder
Theorem, with M. Harada and P. Sole,
Hokkaido Math Journal, Volume 28, 1999, 253-283.
- Type II Self-Dual Codes over Finite Rings and Even
Unimodular Lattices, Journal of Algebraic Combinatorics, Volume 9, Issue 3,
1999, p. 233-250, with T.A. Gulliver and M. Harada.
- Shadow Optimal Self-Dual Codes,
Kyushu Journal of Mathematics, Vol 53, No. 2., 1999, p. 223-237,
with M. Harada.
- Type IV Self-Dual Codes over Rings ,
IEEE-IT, Vol 45, No. 7, November 1999, p. 2345-2360,
(with P. Gaborit, M. Harada,
A. Munemasa, and P. Sole.)
- New Extremal Codes of Length 68 ,
IEEE-IT, Vol 46, No. 6, 2133-2136, September 1999, (with M. Harada).
- Self-Dual Codes and Finite Projective Planes ,
Mathematical Journal of Okayama University, Vol. 40, 1998,(2000).
- Shadow Lattice and Shadow Codes,
Discrete Math, Vol. 219 (2000), 49-64,
(with M. Harada and P. Sole.)
- Optimal Formally Self-Dual Codes over F_5 and
F_7,
AAECC, 10 (2000) No. 3, 227-236, (with M. Harada and T.A. Gulliver.)
- Higher Weights and Binary Self-Dual Codes ,
Electronic Notes in Apllied Mathemtics, April, 2001,
with T. Aaron Gulliver.
-
Shadow Codes over Z_4 , Finite Fields and their Applications,
with M. Harada and P. Sole.
- Higher Weights and Graded Rings for Binary
Self-Dual Codes , with T. Aaron Gulliver and Manabu Oura, to appear
in ADM.
- Codes over Z_4 + w Z_4,
Complex Lattices and Hermitian Modular Forms,
with YoungJu Choie.
- Codes over F_3 + v F_3, with R. Chapman, P. Sole,
and P. Gaborit.
- Complete Joint Weight Enumerators and
Self-Dual Codes , with Young Ju Choie and Haesuk Kim.
- Splitting the Shadow , with A. Bonnecaze,
Y. Choie and P. Sole.
- Generalized Shadows of Codes over Rings .
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