August  2021, 15(3): 507-524. doi: 10.3934/amc.2020079

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

* Corresponding author: J. D. Key

Received  October 2019 Revised  December 2019 Published  April 2020

Fund Project: The second author is supported by the National Research Foundation of South Africa (Grant Numbers 95725 and 106071)

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.

Citation: Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079
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.  Google Scholar

[2]

B. BoseB. BroegY. 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.  Google Scholar

[3]

W. BosmaJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

W. Fish, J. D. Key and E. Mwambene, LCDcodes from products of graphs, In preparation. Google Scholar

[8]

W. FishJ. 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.  Google Scholar

[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.  Google Scholar

[11]

W. C. Huffman, Codes and groups, Handbook of Coding Theory, North-Holland, Amsterdam, 1, 2 (1998), 1345-1440.   Google Scholar

[12]

J. D. KeyT. 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.  Google Scholar

[13]

J. D. KeyT. 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.  Google Scholar

[14]

J. D. KeyT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

C. Kravvaritis, Determinant evaluations for binary circulant matrices, Spec. Matrices, 1 (2013), 187–199. http://dx.doi.org/10.2478/spma-2014-0019.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342.  doi: 10.1016/0012-365X(92)90563-U.  Google Scholar

[22]

J. Schönheim, On coverings, Pacific J. Math., 14 (1964), 1405-1411.  doi: 10.2140/pjm.1964.14.1405.  Google Scholar

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.  Google Scholar

[2]

B. BoseB. BroegY. 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.  Google Scholar

[3]

W. BosmaJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

W. Fish, J. D. Key and E. Mwambene, LCDcodes from products of graphs, In preparation. Google Scholar

[8]

W. FishJ. 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.  Google Scholar

[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.  Google Scholar

[11]

W. C. Huffman, Codes and groups, Handbook of Coding Theory, North-Holland, Amsterdam, 1, 2 (1998), 1345-1440.   Google Scholar

[12]

J. D. KeyT. 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.  Google Scholar

[13]

J. D. KeyT. 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.  Google Scholar

[14]

J. D. KeyT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

C. Kravvaritis, Determinant evaluations for binary circulant matrices, Spec. Matrices, 1 (2013), 187–199. http://dx.doi.org/10.2478/spma-2014-0019.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342.  doi: 10.1016/0012-365X(92)90563-U.  Google Scholar

[22]

J. Schönheim, On coverings, Pacific J. Math., 14 (1964), 1405-1411.  doi: 10.2140/pjm.1964.14.1405.  Google Scholar

Table 1.  Blocks in $ \mathcal{{B}} $$ _m $
$ m $
$ 5 $ $ 0,2 $
$ 7 $ $ 0,0 $ $ 0,3 $ $ 2,2 $
$ 9 $ $ 0,2 $ $ 0,3 $ $ 2,4 $
$ 11 $ $ 0,0 $ $ 0,3 $ $ 0,4 $ $ 2,2 $ $ 2,5 $ $ 4,4 $
$ 13 $ $ 0,2 $ $ 0,3 $ $ 0,6 $ $ 2,4 $ $ 2,5 $ $ 4,6 $
$ 15 $ $ 0,0 $ $ 0,3 $ $ 0,4 $ $ 0,7 $ $ 2,2 $ $ 2,5 $ $ 2,6 $ $ 4,4 $ $ 4,7 $ $ 6,6 $
$ 17 $ $ 0,2 $ $ 0,3 $ $ 0,6 $ $ 0,7 $ $ 2,4 $ $ 2,5 $ $ 2,8 $ $ 4,6 $ $ 4,7 $ $ 6,8 $
$ 19 $ $ 0,0 $ $ 0,3 $ $ 0,4 $ $ 0,7 $ 0, 8 $ 2,2 $ $ 2,5 $ $ 2,6 $ 2, 9 $ 4,4 $ $ 4,7 $ 4, 8 $ 6,6 $ 6, 9 8, 8
$ m $
$ 5 $ $ 0,2 $
$ 7 $ $ 0,0 $ $ 0,3 $ $ 2,2 $
$ 9 $ $ 0,2 $ $ 0,3 $ $ 2,4 $
$ 11 $ $ 0,0 $ $ 0,3 $ $ 0,4 $ $ 2,2 $ $ 2,5 $ $ 4,4 $
$ 13 $ $ 0,2 $ $ 0,3 $ $ 0,6 $ $ 2,4 $ $ 2,5 $ $ 4,6 $
$ 15 $ $ 0,0 $ $ 0,3 $ $ 0,4 $ $ 0,7 $ $ 2,2 $ $ 2,5 $ $ 2,6 $ $ 4,4 $ $ 4,7 $ $ 6,6 $
$ 17 $ $ 0,2 $ $ 0,3 $ $ 0,6 $ $ 0,7 $ $ 2,4 $ $ 2,5 $ $ 2,8 $ $ 4,6 $ $ 4,7 $ $ 6,8 $
$ 19 $ $ 0,0 $ $ 0,3 $ $ 0,4 $ $ 0,7 $ 0, 8 $ 2,2 $ $ 2,5 $ $ 2,6 $ 2, 9 $ 4,4 $ $ 4,7 $ 4, 8 $ 6,6 $ 6, 9 8, 8
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