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Nearly optimal codebooks from generalized Boolean bent functions over $ \mathbb{Z}_{4} $
Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $
1. | Department of Mathematics and Applied Mathematics, University of the Western Cape, 7535 Bellville, South Africa |
2. | Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0028, South Africa |
We examine the binary codes from adjacency matrices of the graph with vertices the nodes of the $ m $-ary $ n $-cube $ Q^m_n $ and with adjacency defined by the Lee metric. For $ n = 2 $ and $ m $ odd, we obtain the parameters of the code and its dual, and show the codes to be $ LCD $. We also find $ s $-PD-sets of size $ s+1 $ for $ s < \frac{m-1}{2} $ for the dual codes, i.e. $ [m^2,2m-1,m]_2 $ codes, when $ n = 2 $ and $ m\ge 5 $ is odd.
References:
[1] |
E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics,103. Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781316529836. |
[2] |
B. Bose, B. Broeg, Y. Kwon and Y. Ashir,
Lee distance and topological properties of $k$-ary $n$-cubes, IEEE Trans. Computers, 44 (1995), 1021-1030.
doi: 10.1109/12.403718. |
[3] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system. I: The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[4] |
J. Cannon, A. Steel and G. White, Linear codes over finite fields, Handbook of Magma Functions, Computational Algebra Group, Department of Mathematics, University of Sydney, (2006), 3951–4023. http://magma.maths.usyd.edu.au/magma. Google Scholar |
[5] |
K. Day and A. E. Al Ayyoub,
Fault diameter of $k$-ary $n$-cube networks, IEEE Trans. Parallel and Distributed Systems, 8 (1997), 903-907.
doi: 10.1109/71.615436. |
[6] |
W. Fish,
Binary codes and permutation decoding sets from the graph products of cycles, Appl. Algebra Engrg. Comm. Comput., 28 (2017), 369-389.
doi: 10.1007/s00200-016-0310-y. |
[7] |
W. Fish, J. D. Key and E. Mwambene, LCDcodes from products of graphs, In preparation. Google Scholar |
[8] |
W. Fish, J. D. Key and E. Mwambene,
Codes, designs and groups from the Hamming graphs, J. Combin. Inform. System Sci., 34 (2009), 169-182.
doi: 10.1016/j.disc.2008.09.024. |
[9] |
W. Fish, Codes from Uniform Subset Graphs and Cycle Products, PhD thesis, University of the Western Cape, 2007. Google Scholar |
[10] |
D. M. Gordon,
Minimal permutation sets for decoding the binary Golay codes, IEEE Trans. Inform. Theory, 28 (1982), 541-543.
doi: 10.1109/TIT.1982.1056504. |
[11] |
W. C. Huffman,
Codes and groups, Handbook of Coding Theory, North-Holland, Amsterdam, 1, 2 (1998), 1345-1440.
|
[12] |
J. D. Key, T. P. McDonough and V. C. Mavron,
Partial permutation decoding for codes from finite planes, European J. Combin., 26 (2005), 665-682.
doi: 10.1016/j.ejc.2004.04.007. |
[13] |
J. D. Key, T. P. McDonough and V. C. Mavron,
Information sets and partial permutation decoding for codes from finite geometries, Finite Fields Appl., 12 (2006), 232-247.
doi: 10.1016/j.ffa.2005.05.007. |
[14] |
J. D. Key, T. P. Mc{D}onough and V. C. Mavron,
Improved partial permutation decoding for Reed-Muller codes, Discrete Math., 340 (2017), 722-728.
doi: 10.1016/j.disc.2016.11.031. |
[15] |
J. D. Key and B. G. Rodrigues,
LCD codes from adjacency matrices of graphs, Appl. Algebra Engrg. Comm. Comput., 29 (2018), 227-244.
doi: 10.1007/s00200-017-0339-6. |
[16] |
J. D. Key and B. G. Rodrigues,
Special $LCD$ codes from {P}eisert and generalized Peisert graphs, Graphs Combin., 35 (2019), 633-652.
doi: 10.1007/s00373-019-02019-0. |
[17] |
C. Kravvaritis, Determinant evaluations for binary circulant matrices, Spec. Matrices, 1 (2013), 187–199. http://dx.doi.org/10.2478/spma-2014-0019. |
[18] |
H.-J. Kroll and R. Vincenti,
PD-sets related to the codes of some classical varieties, Discrete Math., 301 (2005), 89-105.
doi: 10.1016/j.disc.2004.11.020. |
[19] |
F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964), 485-505. Google Scholar |
[20] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
[21] |
J. L. Massey,
Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342.
doi: 10.1016/0012-365X(92)90563-U. |
[22] |
J. Schönheim,
On coverings, Pacific J. Math., 14 (1964), 1405-1411.
doi: 10.2140/pjm.1964.14.1405. |
show all references
References:
[1] |
E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics,103. Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781316529836. |
[2] |
B. Bose, B. Broeg, Y. Kwon and Y. Ashir,
Lee distance and topological properties of $k$-ary $n$-cubes, IEEE Trans. Computers, 44 (1995), 1021-1030.
doi: 10.1109/12.403718. |
[3] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system. I: The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[4] |
J. Cannon, A. Steel and G. White, Linear codes over finite fields, Handbook of Magma Functions, Computational Algebra Group, Department of Mathematics, University of Sydney, (2006), 3951–4023. http://magma.maths.usyd.edu.au/magma. Google Scholar |
[5] |
K. Day and A. E. Al Ayyoub,
Fault diameter of $k$-ary $n$-cube networks, IEEE Trans. Parallel and Distributed Systems, 8 (1997), 903-907.
doi: 10.1109/71.615436. |
[6] |
W. Fish,
Binary codes and permutation decoding sets from the graph products of cycles, Appl. Algebra Engrg. Comm. Comput., 28 (2017), 369-389.
doi: 10.1007/s00200-016-0310-y. |
[7] |
W. Fish, J. D. Key and E. Mwambene, LCDcodes from products of graphs, In preparation. Google Scholar |
[8] |
W. Fish, J. D. Key and E. Mwambene,
Codes, designs and groups from the Hamming graphs, J. Combin. Inform. System Sci., 34 (2009), 169-182.
doi: 10.1016/j.disc.2008.09.024. |
[9] |
W. Fish, Codes from Uniform Subset Graphs and Cycle Products, PhD thesis, University of the Western Cape, 2007. Google Scholar |
[10] |
D. M. Gordon,
Minimal permutation sets for decoding the binary Golay codes, IEEE Trans. Inform. Theory, 28 (1982), 541-543.
doi: 10.1109/TIT.1982.1056504. |
[11] |
W. C. Huffman,
Codes and groups, Handbook of Coding Theory, North-Holland, Amsterdam, 1, 2 (1998), 1345-1440.
|
[12] |
J. D. Key, T. P. McDonough and V. C. Mavron,
Partial permutation decoding for codes from finite planes, European J. Combin., 26 (2005), 665-682.
doi: 10.1016/j.ejc.2004.04.007. |
[13] |
J. D. Key, T. P. McDonough and V. C. Mavron,
Information sets and partial permutation decoding for codes from finite geometries, Finite Fields Appl., 12 (2006), 232-247.
doi: 10.1016/j.ffa.2005.05.007. |
[14] |
J. D. Key, T. P. Mc{D}onough and V. C. Mavron,
Improved partial permutation decoding for Reed-Muller codes, Discrete Math., 340 (2017), 722-728.
doi: 10.1016/j.disc.2016.11.031. |
[15] |
J. D. Key and B. G. Rodrigues,
LCD codes from adjacency matrices of graphs, Appl. Algebra Engrg. Comm. Comput., 29 (2018), 227-244.
doi: 10.1007/s00200-017-0339-6. |
[16] |
J. D. Key and B. G. Rodrigues,
Special $LCD$ codes from {P}eisert and generalized Peisert graphs, Graphs Combin., 35 (2019), 633-652.
doi: 10.1007/s00373-019-02019-0. |
[17] |
C. Kravvaritis, Determinant evaluations for binary circulant matrices, Spec. Matrices, 1 (2013), 187–199. http://dx.doi.org/10.2478/spma-2014-0019. |
[18] |
H.-J. Kroll and R. Vincenti,
PD-sets related to the codes of some classical varieties, Discrete Math., 301 (2005), 89-105.
doi: 10.1016/j.disc.2004.11.020. |
[19] |
F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964), 485-505. Google Scholar |
[20] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
[21] |
J. L. Massey,
Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342.
doi: 10.1016/0012-365X(92)90563-U. |
[22] |
J. Schönheim,
On coverings, Pacific J. Math., 14 (1964), 1405-1411.
doi: 10.2140/pjm.1964.14.1405. |
0, 8 | 2, 9 | 4, 8 | 6, 9 | 8, 8 |
0, 8 | 2, 9 | 4, 8 | 6, 9 | 8, 8 |
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