American Institute of Mathematical Sciences

Advanced
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.  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.  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.   Google Scholar

[4]

A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identites,, in, 3 (1971), 211.   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

V. Pless, W. C. Huffman and R. A. Brualdi (eds.), "Handbook of Coding Theory,", North-Holland, (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.  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.  doi: 10.1109/18.651000.  Google Scholar

[11]

W. Scharlau, Quadratic and Hermitian forms,, in, (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.  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.  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.   Google Scholar

[4]

A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identites,, in, 3 (1971), 211.   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

V. Pless, W. C. Huffman and R. A. Brualdi (eds.), "Handbook of Coding Theory,", North-Holland, (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.  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.  doi: 10.1109/18.651000.  Google Scholar

[11]

W. Scharlau, Quadratic and Hermitian forms,, in, (1985).   Google Scholar

[12]

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

[1]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[2]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003
[3]

Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161
[4]

Crnković Dean, Vedrana Mikulić Crnković, Bernardo G. Rodrigues. On self-orthogonal designs and codes related to Held's simple group. Advances in Mathematics of Communications, 2018, 12 (3) : 607-628. doi: 10.3934/amc.2018036
[5]

Leetika Kathuria, Madhu Raka. Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless. Advances in Mathematics of Communications, 2012, 6 (4) : 499-503. doi: 10.3934/amc.2012.6.499
[6]

Liren Lin, Hongwei Liu, Bocong Chen. Existence conditions for self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (1) : 1-7. doi: 10.3934/amc.2015.9.1
[7]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261
[8]

Amita Sahni, Poonam Trama Sehgal. Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (4) : 437-447. doi: 10.3934/amc.2015.9.437
[9]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[10]

Dean Crnković, Marija Maksimović, Bernardo Gabriel Rodrigues, Sanja Rukavina. Self-orthogonal codes from the strongly regular graphs on up to 40 vertices. Advances in Mathematics of Communications, 2016, 10 (3) : 555-582. doi: 10.3934/amc.2016026
[11]

Xia Li, Feng Cheng, Chunming Tang, Zhengchun Zhou. Some classes of LCD codes and self-orthogonal codes over finite fields. Advances in Mathematics of Communications, 2019, 13 (2) : 267-280. doi: 10.3934/amc.2019018
[12]

Dean Crnković, Ronan Egan, Andrea Švob. Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020032
[13]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1
[14]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
[15]

Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027
[16]

Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275
[17]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399
[18]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
[19]

Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503
[20]

Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences