Abstract
Studies of the p-ary codes from the adjacency matrices of uniform subset graphs \(\varGamma (n,k,r)\) and their reflexive associates have shown that a particular family of codes defined on the subsets are intimately related to the codes from these graphs. We describe these codes here and examine their relation to some particular classes of uniform subset graphs. In particular we include a complete analysis of the p-ary codes from \(\varGamma (n,3,r)\) for \(p\ge 5\), thus extending earlier results for \(p=2,3\).
Similar content being viewed by others
Notes
The authors thank T.P. McDonough for this observation.
References
Araujo, J., Bratten, T.: On the spectrum of the Johnson graphs \({J}(n,k,r)\). In: Proceedings of the XXIIrd “Dr. Antonio A. R. Monteiro” Congress, pp. 57–62 (2016). Univ. Nac. Sur Dep. Mat. Inst. Mat., Baha Blanca, 2015
Assmus Jr., E.F., Key, J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992). Cambridge Tracts in Mathematics, Vol. 103 (Second printing with corrections, 1993)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24(3/4), 235–265 (1997)
Brouwer, A., Haemers, W.: Association schemes. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, Chapter 15, Vol. 1, pp. 749–771. Elsevier; MIT, Cambridge (1995)
Cannon, J., Steel, A., White, G.: Linear codes over finite fields. In: Cannon, J., Bosma, W. (eds.) Handbook of Magma Functions, pp. 3951–4023. Computational Algebra Group, Department of Mathematics, University of Sydney (2006). V2.13. http://magma.maths.usyd.edu.au/magma
Delsarte, P.: An Algebraic Approach to the Association Schemes of Coding Theory. Tech. rep., Philips Research Laboratorie (1973). Philips Research Reports, Supplement No. 10
Delsarte, P., Levenshtein, V.I.: Association schemes and coding theory. IEEE Trans. Inf. Theory 44, 2477–2504 (1998)
Fish, W.: Codes from uniform subset graphs and cycle products. Ph.D. thesis, University of the Western Cape (2007)
Fish, W.: Binary codes and partial permutation decoding sets from the Johnson graphs. Graphs Comb. 31, 1381–1396 (2015)
Fish, W., Fray, R., Mwambene, E.: Binary codes and partial permutation decoding sets from the odd graphs. Cent. Eur. J. Math. 12(9), 1362–1371 (2014)
Fish, W., Key, J.D., Mwambene, E.: Ternary codes from reflexive graphs on \(3\)-sets. Appl. Algebra Eng. Commun. Comput. 25, 363–382 (2014)
Fish, W., Key, J.D., Mwambene, E.: Binary codes from reflexive graphs on \(3\)-sets. Adv. Math. Commun. 9, 211–232 (2015)
Huffman, W.C.: Codes and groups. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, vol. 2, Part 2, Chapter 17, pp. 1345–1440. Elsevier, Amsterdam (1998)
Key, J.D., Moori, J., Rodrigues, B.G.: Binary codes from graphs on triples. Discret. Math. 282(1–3), 171–182 (2004)
Key, J.D., Moori, J., Rodrigues, B.G.: Ternary codes from graphs on triples. Discret. Math. 309, 4663–4681 (2009)
Key, J.D., Rodrigues, B.G.: \(LCD\) codes from adjacency matrices of graphs. (To appear) Appl. Algebra Eng. Commun. Comput. https://doi.org/10.1007/s00200-017-0339-6
Kirkman, T.P.: On a problem in combinations. Camb. Dublin Math. J. 2, 191–204 (1847)
Krebs, M., Shaheen, A.: On the spectra of Johnson graphs. Electron. J. Linear Algebra 17, 154–167 (2008)
Massey, J.L.: Linear codes with complementary duals. Discret. Math. 106(107), 337–342 (1992)
Moore, E.H.: Concerning triple systems. Math. Ann. 43, 271–285 (1893)
Peeters, R.: On the \(p\)-ranks of the adjacency matrices of distance-regular graphs. J. Algebraic Comb. 15, 127–149 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fish, W., Key, J.D. & Mwambene, E. Codes from Adjacency Matrices of Uniform Subset Graphs. Graphs and Combinatorics 34, 163–192 (2018). https://doi.org/10.1007/s00373-017-1862-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-017-1862-8