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November  2010, 4(4): 579-596. doi: 10.3934/amc.2010.4.579

On dual extremal maximal self-orthogonal codes of Type I-IV

Annika Meyer 1,

1. 

Lehrstuhl D für Mathematik, RWTH Aachen University, Templergraben 64, 52062 Aachen, Germany

Received  February 2010 Revised  September 2010 Published  November 2010

For a Type $T \in${I, II, III, IV} of codes over finite fields and length $N$ where there exists no self-dual Type $T$ code of length $N$, upper bounds on the minimum weight of the dual code of a self-orthogonal Type $T$ code of length $N$ are given, allowing the notion of dual extremal codes. It is proven that for $T \in${II, III, IV} the Hamming weight enumerator of a dual extremal maximal self-orthogonal Type $T$ code of a given length is unique.
Keywords: self-orthogonal code, weight distribution, extremality., Linear code.
Mathematics Subject Classification: 11T7.
Citation: Annika Meyer. On dual extremal maximal self-orthogonal codes of Type I-IV. Advances in Mathematics of Communications, 2010, 4 (4) : 579-596. doi: 10.3934/amc.2010.4.579
References:
[1]

C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows, J. Combin. Theory, 105 (2004), 15-34. doi: 10.1016/j.jcta.2003.09.003.  Google Scholar

[2]

J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931.  Google Scholar

[3]

P. Gaborit, A bound for certain s-extremal lattices and codes, Arch. Math. (Basel), 89 (2007), 143-151.  Google Scholar

[4]

A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identites, in "Actes, Congr. Int. Math.'' (ed. P. Gauthier-Villars), 3 (1971), 211-215.  Google Scholar

[5]

W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes," Cambridge University Press, Cambridge, 2003.  Google Scholar

[6]

G. Nebe, E. M. Rains and N. J. A. Sloane, "Self-Dual Codes and Invariant Theory," Springer, Berlin, 2006.  Google Scholar

[7]

M. Ozeki, On intersection properties of extremal ternary codes, J. Math. Industry, 1 (2009), 105-121.  Google Scholar

[8]

V. Pless, W. C. Huffman and R. A. Brualdi (eds.), "Handbook of Coding Theory," North-Holland, Amsterdam, 1998.  Google Scholar

[9]

V. Pless, N. J. A. Sloane and H. N. Ward, Ternary codes of minimum weight 6, and the classification of length 20, IEEE Trans. Inform. Theory, 26 (1980), 305-316. doi: 10.1109/TIT.1980.1056195.  Google Scholar

[10]

E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.  Google Scholar

[11]

W. Scharlau, Quadratic and Hermitian forms, in "Grundlehren der Mathematischen Wissenschaften," Springer-Verlag, 1985.  Google Scholar

[12]

N. J. A. Sloane, Gleason's Theorem on self-dual codes and its generalizations,, preprint, ().   Google Scholar

show all references

References:
[1]

C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows, J. Combin. Theory, 105 (2004), 15-34. doi: 10.1016/j.jcta.2003.09.003.  Google Scholar

[2]

J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931.  Google Scholar

[3]

P. Gaborit, A bound for certain s-extremal lattices and codes, Arch. Math. (Basel), 89 (2007), 143-151.  Google Scholar

[4]

A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identites, in "Actes, Congr. Int. Math.'' (ed. P. Gauthier-Villars), 3 (1971), 211-215.  Google Scholar

[5]

W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes," Cambridge University Press, Cambridge, 2003.  Google Scholar

[6]

G. Nebe, E. M. Rains and N. J. A. Sloane, "Self-Dual Codes and Invariant Theory," Springer, Berlin, 2006.  Google Scholar

[7]

M. Ozeki, On intersection properties of extremal ternary codes, J. Math. Industry, 1 (2009), 105-121.  Google Scholar

[8]

V. Pless, W. C. Huffman and R. A. Brualdi (eds.), "Handbook of Coding Theory," North-Holland, Amsterdam, 1998.  Google Scholar

[9]

V. Pless, N. J. A. Sloane and H. N. Ward, Ternary codes of minimum weight 6, and the classification of length 20, IEEE Trans. Inform. Theory, 26 (1980), 305-316. doi: 10.1109/TIT.1980.1056195.  Google Scholar

[10]

E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.  Google Scholar

[11]

W. Scharlau, Quadratic and Hermitian forms, in "Grundlehren der Mathematischen Wissenschaften," Springer-Verlag, 1985.  Google Scholar

[12]

N. J. A. Sloane, Gleason's Theorem on self-dual codes and its generalizations,, preprint, ().   Google Scholar

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