<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 9 Issue 2" %>
Extending MDS Codes
T.L. Alderson
Department of Mathematical Sciences, University of New Brunswick, Saint John, Canada
talderso@unb.ca
Annals of Combinatorics 9 (2) p.125-135 June, 2005
AMS Subject Classification: 94B25, 51E21, 05B15
Abstract:
A q-ary (n, k)-MDS code, linear or not, satisfies . A code meeting this bound is said to have maximum length. Using purely combinatorial methods we show that an MDS code with can be uniquely extended to a full length code if and only if q is even. This result is best possible in the sense that there is, for example, a non-extendable 4-ary (5, 4)-MDS code. It may be that the proof of our result is as interesting as the result itself. We provide a simple necessary and sufficient condition (property P) for code extendability. In future work, this condition might be suitably modified to give an extendability condition for arbitrary (shorter) MDS codes.
Keywords: MDS code, Latin hypercube, code extension

References:

1. T.L. Alderson, On MDS codes and Bruen-Silverman codes, Ph.D. Thesis, University of Western Ontario, 2002.

2. R. Baer, Nets and groups, Trans. Amer. Math. Soc. 46 (1939) 110--141.

3. L.M. Batten, Combinatorics of Finite Geometries, Cambridge Univ. Press, 1986.

4. A. Beutelspacher and K. Metsch, Embedding finite linear spaces in projective planes, Ann. Discrete Math. 30 (1984) 39--56.

5. A. Beutelspacher and K. Metsch, Embedding finite linear spaces in projective planes II, Discr. Math. 66 (1987) 219--230.

6. R.C. Bose, S.S. Shrikhande, and E.T. Parker, Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler's conjecture, Canad. J. Math. 12 (1960) 189--203.

7. R.C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math. 13 (1963) 389--419.

8. R.H. Bruck, Finite nets I, numerical invariants, Canad J. Math. 3 (1951) 94--107.

9. R.H. Bruck, Finite nets II, uniqueness and embedding, Pacific J. Math. 13 (1963) 421--457.

10. R.H. Bruck and H.J. Ryser, The nonexistence of certain finite projective planes, Canad J. Math. 1 (1949) 88--93.

11. A.A. Bruen, Partial spreads and replacable nets, Canad J. Math. 23 (1971) 381--391.

12. A.A. Bruen, Unembeddable nets of small deficiency, Pacific J. Math. 43 (1972) 51--54.

13. A.A. Bruen, Collineations and extensions of translation nets, Math. Z. 145 (1975) 243--249.

14. A.A. Bruen and R. Silverman, On the nonexistence of certain MDS codes and projective planes, Math. Z. 183 (1983) 171--175.

15. A.A. Bruen and R. Silverman, On extendable planes, MDS codes and hyperovals in PG(2,q), q = 2t , Geom. Dedicata 28 (1988) 31--43.

16. A.A. Bruen, J.A. Thas, and A. Blokhuis, MDS codes and arcs in projective spaces I, C.R. Math. Rep. Acad. Sci. Canada 10 (1988) 225--230.

17. A.A. Bruen, J.A. Thas, and A. Blokhuis, MDS codes and arcs in projective spaces II, C.R. Math. Rep. Acad. Sci. Canada 10 (1988) 233--235.

18. J.W.P. Hirschfeld, Complete arcs, Discrete Math. 174 (1-3) (1997) 177--184.

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

20. K. Metsch, Improvement of Bruck's completion theorem, Des. Codes Crypogr. 1 (1991) 99--116.

21. C.R. Rao, Hypercubes of strength d leading to confounded designs in factorial experiments, Bull. Calcutta Math. Soc. 38 (1946) 67--78.

22. R. Silverman, A metrization for power-sets with applications to combinatorial analysis, Canad. J. Math. 12 (1960) 158--176.

23. R. Silverman and C. Maneri, A vector space packing problem, J. Algebra 4 (1966) 321--330.

24.L. Storme and J.A. Thas, M.D.S. codes and k-arcs in PG(n;q) with q even: an improvement of the bounds of Bruen, Thas and Blokhuis, J. Combin. Theory Ser. A 62 (1) (1993) 139--154.

25. S.B. Wicker and V. Bhargava, Reed-Solomon Codes and Their Applications, IEEE Press, NewYork, 1994.