August  2016, 10(3): 489-498. doi: 10.3934/amc.2016020

Further results on the classification of MDS codes

1. 

Department of Communications and Networking, School of Electrical Engineering, Aalto University, P.O. Box 13000, 00076 Aalto, Finland

Received  April 2015 Revised  June 2016 Published  August 2016

An unrestricted $q$-ary maximum distance separable (MDS) code $C$ with length $n$ over an alphabet $\mathcal{A}$ (of size $q$) is a set of $q^k$ codewords that are elements of $\mathcal{A}^n$, such that the smallest Hamming distance between two distinct codewords in $C$ is $d=n-k+1$. Sets of mutually orthogonal Latin squares of orders $q\leq 9$, corresponding to $q$-ary MDS codes of size $q^2$, and $q$-ary one-error-correcting MDS codes for $q\leq 8$ have been classified in earlier studies. These results are used here to complete the classification of all $7$-ary and $8$-ary MDS codes with $d\geq 3$ using a computer search.
Citation: Janne I. Kokkala, Patric R. J.  Östergård. Further results on the classification of MDS codes. Advances in Mathematics of Communications, 2016, 10 (3) : 489-498. doi: 10.3934/amc.2016020
References:
[1]

T. L. Alderson, $(6,3)$-MDS codes over an alphabet of size 4,, Des. Codes Cryptogr., 38 (2006), 31. doi: 10.1007/s10623-004-5659-4.

[2]

T. L. Alderson, A. A. Bruen and R. Silverman, Maximum distance separable codes and arcs in projective spaces,, J. Combin. Theory Ser. A, 114 (2007), 1101. doi: 10.1016/j.jcta.2006.11.005.

[3]

T. L. Alderson and A. Gács, On the maximality of linear codes,, Des. Codes Cryptogr., 53 (2009), 59. doi: 10.1007/s10623-009-9293-z.

[4]

S. Ball, On sets of vectors of a finite vector space in which every subset of basis size is a basis,, J. Eur. Math. Soc., 14 (2012), 733. doi: 10.4171/JEMS/316.

[5]

S. Ball and J. De Beule, On sets of vectors of a finite vector space in which every subset of basis size is a basis II,, Des. Codes Cryptogr., 65 (2012), 5. doi: 10.1007/s10623-012-9658-6.

[6]

A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert and A. Wassermann, Error-Correcting Linear Codes: Classification by Isometry and Applications,, Springer, (2006).

[7]

J. Egan and I. M. Wanless, Enumeration of MOLS of small order,, Math. Comp., 85 (2016), 799. doi: 10.1090/mcom/3010.

[8]

P. Kaski and P. R. J. Östergård, Classification Algorithms for Codes and Designs,, Springer, (2006).

[9]

P. Kaski and O. Pottonen, Libexact User's Guide, version 1.0, Technical Reports TR 2008-1,, Helsinki Inst. Inform. Techn., (2008).

[10]

G. Kéri, Types of superregular matrices and the number of $n$-arcs and complete $n$-arcs in $PG(r, q)$,, J. Combin. Des., 14 (2006), 363. doi: 10.1002/jcd.20091.

[11]

J. I. Kokkala, D. S. Krotov and P. R. J. Östergård, On the classification of MDS codes,, IEEE Trans. Inf. Theory, 61 (2015), 6485. doi: 10.1109/TIT.2015.2488659.

[12]

J. I. Kokkala and P. R. J. Östergård, Classification of Graeco-Latin cubes,, J. Combin. Des., 23 (2015), 509. doi: 10.1002/jcd.21400.

[13]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, North-Holland, (1977).

[14]

B. M. Maenhaut and I. M. Wanless, Atomic Latin squares of order eleven,, J. Combin. Des., 12 (2004), 12. doi: 10.1002/jcd.10064.

[15]

B. D. McKay, A. Meynert and W. Myrvold, Small Latin squares, quasigroups, and loops,, J. Combin. Des., 15 (2007), 98. doi: 10.1002/jcd.20105.

[16]

B. D. McKay and A. Piperno, Practical graph isomorphism, II,, J. Symbolic Comput., 60 (2014), 94. doi: 10.1016/j.jsc.2013.09.003.

[17]

B. D. McKay and I. M. Wanless, A census of small Latin hypercubes,, SIAM J. Discrete Math., 22 (2008), 719. doi: 10.1137/070693874.

[18]

S. Niskanen and P. R. J. Östergård, Cliquer User's guide, Version 1.0, Technical Report T48,, Communications Laboratory, (2003).

[19]

P. R. J. Östergård, Constructing combinatorial objects via cliques,, in Surveys in Combinatorics 2005 (ed. B. S. Webb), (2005), 57. doi: 10.1017/CBO9780511734885.004.

[20]

E. T. Parker, Computer investigation of orthogonal Latin squares of order ten,, in Proc. Sympos. Appl. Math., (1963), 73.

[21]

V. N. Potapov and D. S. Krotov, On the number of $n$-ary quasigroups of finite order,, Discrete Math. Appl., 21 (2011), 575. doi: 10.1016/j.disc.2010.09.023.

[22]

B. Segre, Curve razionali normali e $k$-archi negli spazi finiti,, Ann. Mat. Pura Appl., 39 (1955), 357. doi: 10.1007/BF02410779.

show all references

References:
[1]

T. L. Alderson, $(6,3)$-MDS codes over an alphabet of size 4,, Des. Codes Cryptogr., 38 (2006), 31. doi: 10.1007/s10623-004-5659-4.

[2]

T. L. Alderson, A. A. Bruen and R. Silverman, Maximum distance separable codes and arcs in projective spaces,, J. Combin. Theory Ser. A, 114 (2007), 1101. doi: 10.1016/j.jcta.2006.11.005.

[3]

T. L. Alderson and A. Gács, On the maximality of linear codes,, Des. Codes Cryptogr., 53 (2009), 59. doi: 10.1007/s10623-009-9293-z.

[4]

S. Ball, On sets of vectors of a finite vector space in which every subset of basis size is a basis,, J. Eur. Math. Soc., 14 (2012), 733. doi: 10.4171/JEMS/316.

[5]

S. Ball and J. De Beule, On sets of vectors of a finite vector space in which every subset of basis size is a basis II,, Des. Codes Cryptogr., 65 (2012), 5. doi: 10.1007/s10623-012-9658-6.

[6]

A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert and A. Wassermann, Error-Correcting Linear Codes: Classification by Isometry and Applications,, Springer, (2006).

[7]

J. Egan and I. M. Wanless, Enumeration of MOLS of small order,, Math. Comp., 85 (2016), 799. doi: 10.1090/mcom/3010.

[8]

P. Kaski and P. R. J. Östergård, Classification Algorithms for Codes and Designs,, Springer, (2006).

[9]

P. Kaski and O. Pottonen, Libexact User's Guide, version 1.0, Technical Reports TR 2008-1,, Helsinki Inst. Inform. Techn., (2008).

[10]

G. Kéri, Types of superregular matrices and the number of $n$-arcs and complete $n$-arcs in $PG(r, q)$,, J. Combin. Des., 14 (2006), 363. doi: 10.1002/jcd.20091.

[11]

J. I. Kokkala, D. S. Krotov and P. R. J. Östergård, On the classification of MDS codes,, IEEE Trans. Inf. Theory, 61 (2015), 6485. doi: 10.1109/TIT.2015.2488659.

[12]

J. I. Kokkala and P. R. J. Östergård, Classification of Graeco-Latin cubes,, J. Combin. Des., 23 (2015), 509. doi: 10.1002/jcd.21400.

[13]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, North-Holland, (1977).

[14]

B. M. Maenhaut and I. M. Wanless, Atomic Latin squares of order eleven,, J. Combin. Des., 12 (2004), 12. doi: 10.1002/jcd.10064.

[15]

B. D. McKay, A. Meynert and W. Myrvold, Small Latin squares, quasigroups, and loops,, J. Combin. Des., 15 (2007), 98. doi: 10.1002/jcd.20105.

[16]

B. D. McKay and A. Piperno, Practical graph isomorphism, II,, J. Symbolic Comput., 60 (2014), 94. doi: 10.1016/j.jsc.2013.09.003.

[17]

B. D. McKay and I. M. Wanless, A census of small Latin hypercubes,, SIAM J. Discrete Math., 22 (2008), 719. doi: 10.1137/070693874.

[18]

S. Niskanen and P. R. J. Östergård, Cliquer User's guide, Version 1.0, Technical Report T48,, Communications Laboratory, (2003).

[19]

P. R. J. Östergård, Constructing combinatorial objects via cliques,, in Surveys in Combinatorics 2005 (ed. B. S. Webb), (2005), 57. doi: 10.1017/CBO9780511734885.004.

[20]

E. T. Parker, Computer investigation of orthogonal Latin squares of order ten,, in Proc. Sympos. Appl. Math., (1963), 73.

[21]

V. N. Potapov and D. S. Krotov, On the number of $n$-ary quasigroups of finite order,, Discrete Math. Appl., 21 (2011), 575. doi: 10.1016/j.disc.2010.09.023.

[22]

B. Segre, Curve razionali normali e $k$-archi negli spazi finiti,, Ann. Mat. Pura Appl., 39 (1955), 357. doi: 10.1007/BF02410779.

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