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Maximum distance separable codes and arcs in projective spaces. (English) Zbl 1126.94017
An $$(n,k,q)$$-MDS code $$C$$ is a collection of $$q^k$$ distinct $$n$$-tuples, called codewords, over an alphabet $${\mathcal A}$$ of size $$q$$, satisfying the following condition: no two codewords of $$C$$ agree in as many as $$k$$ coordinate positions. The importance of these MDS codes is that they satisfy the Singleton bound of coding theory, that is, they are codes of length $$n$$, containing $$q^k$$ codewords, and whose minimal distance $$d$$ is equal to $$d=n-k+1$$.
Linear $$[n,k,n-k+1]$$-MDS codes over the finite field of order $$q$$ are equivalent to $$n$$-arcs in PG$$(k-1,q)$$ and to $$n$$-arcs in PG$$(n-k-1,q)$$. This geometrical link to arcs has made it possible to prove many results on linear MDS codes. In particular, great attention has been paid to the problem of the extendability of $$n$$-arcs in PG$$(k-1,q)$$ to $$(n+1)$$-arcs in PG$$(k-1,q)$$; in this way studying the problem of the extendability of the corresponding $$[n,k,n-k+1]$$-MDS codes to $$[n+1,k,n-k+2]$$-MDS codes.
In some cases, the non-extendability of linear $$[n,k,n-k+1]$$-MDS codes to linear $$[n+1,k,n-k+2]$$-MDS codes is known. But could these codes be extended to non-linear MDS codes of length $$n+1$$?
The authors contribute to this particular extendability problem. They obtain new results by using new geometrical links. The new links are with Rédei-type blocking sets.
Consider an affine plane $$A$$ of order $$q$$, with line $$\ell$$ at infinity. Let $$\pi$$ be the projective plane defined by $$A$$ and $$\ell$$. A Rédei-type blocking set of $$\pi$$, w.r.t. the line $$\ell$$, is a set $$B$$ consisting of $$q$$ points of $$A$$, together with the intersection points of all secants to $$A$$ with the line $$\ell$$.
These Rédei-type blocking sets in PG$$(2,q)$$ have been studied in great detail by A. Blokhuis, S. Ball, A.E. Brouwer, L. Storme and T. Szőnyi [J. Comb. Theory, Ser. A 86, No. 1, 187–196 (1999; Zbl 0945.51002)] and S. Ball [J. Comb. Theory, Ser. A 104, No. 2, 341–350 (2003; Zbl 1045.51004)]. Let $${\mathcal P}_q>1$$ be the smallest size for the intersection $$B\cap \ell$$ of a Rédei-type blocking set $$B$$, different from a line, w.r.t. $$\ell$$.
To illustrate the link between the extendability problem of linear MDS codes and Rédei-type blocking sets of PG$$(2,q)$$, we mention the following result: Let $$C$$ be a linear $$[n,3,n-2]$$-MDS code, with $$n>q+2-{\mathcal P}_q$$. Then any arbitrary extension of $$C$$ to an MDS code of length $$n+1$$ must be linear.
We also wish to mention with respect to Section 7.1 the results of L. Storme and J. A. Thas [J. Comb. Theory, Ser. A 62, No. 1, 139–154 (1993; Zbl 0771.51006)] on arcs in PG$$(N,q)$$, $$q$$ even.
Reviewer: Leo Storme (Gent)

##### MSC:
 94B25 Combinatorial codes 05B25 Combinatorial aspects of finite geometries 51E14 Finite partial geometries (general), nets, partial spreads 51E20 Combinatorial structures in finite projective spaces 51E21 Blocking sets, ovals, $$k$$-arcs
##### Keywords:
MDS codes; code extensions; linear codes; arcs; dual arcs; complete arcs
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##### References:
  Alderson, T.L., Extending MDS codes, Ann. comb., 9, 2, 125-135, (2005) · Zbl 1112.51002  T.L. Alderson, On MDS codes and Bruen-Silverman codes, PhD thesis, University of Western Ontario, 2002  Ball, S., The number of directions determined by a function over a finite field, J. combin. theory ser. A, 104, 2, 341-350, (2003) · Zbl 1045.51004  Blokhuis, A.; Ball, S.; Brouwer, A.E.; Storme, L.; Szőnyi, T., On the number of slopes of the graph of a function defined on a finite field, J. combin. theory ser. A, 86, 1, 187-196, (1999) · Zbl 0945.51002  Blokhuis, A.; Bruen, A.A.; Thas, J.A., Arcs in $$\mathit{PG}(n, q)$$, MDS-codes and three fundamental problems of B. segre—some extensions, Geom. dedicata, 35, 1-3, 1-11, (1990) · Zbl 0709.51013  Bruen, A.A., Collineations and extensions of translation nets, Math. Z., 145, 3, 243-249, (1975) · Zbl 0297.50017  Bruen, A.A., Nuclei of sets of $$q + 1$$ points in $$\mathit{PG}(2, q)$$ and blocking sets of Rédei type, J. combin. theory ser. A, 55, 1, 130-132, (1990) · Zbl 0706.51011  Bruen, A.A.; Forcinito, Mario A., Cryptography, information theory, and error-correction, Wiley – intersci. ser. discrete math. optim., (2005), Wiley-Interscience Hoboken, NJ, A handbook for the 21st century · Zbl 1071.94001  Bruen, A.A.; Levinger, B., A theorem on permutations of a finite field, Canad. J. math., 25, 1060-1065, (1973) · Zbl 0269.12011  Bruen, A.A.; Silverman, R., On extendable planes, M.D.S. codes and hyperovals in $$\mathit{PG}(2, q)$$, $$q = 2^t$$, Geom. dedicata, 28, 1, 31-43, (1988) · Zbl 0661.94016  Bruen, A.A.; Thas, J.A.; Blokhuis, A., On M.D.S. codes, arcs in $$\mathit{PG}(n, q)$$ with q even, and a solution of three fundamental problems of B. Segre, Invent. math., 92, 3, 441-459, (1988) · Zbl 0654.94014  Calderbank, A.R.; Shor, Peter W., Good quantum error-correcting codes exist, Phys. rev. A, 54, 1098, (1996)  Casse, Louis Reynolds Antoine, A solution to beniamino Segre’s “problem $$I_{r, q}$$” for q even, Atti accad. naz. lincei rend. cl. sci. fis. mat. natur. (8), 46, 13-20, (1969) · Zbl 0176.17901  Hirschfeld, J.W.P., Projective geometries over finite fields, Oxford math. monogr., (1998), Clarendon Press, Oxford Univ. Press New York · Zbl 0899.51002  Lovász, L.; Schrijver, A., Remarks on a theorem of Rédei, Studia sci. math. hungar., 16, 3-4, 449-454, (1983) · Zbl 0535.51009  MacWilliams, F.J.; Sloane, N.J.A., The theory of error-correcting codes. I, North-holland math. library, vol. 16, (1977), North-Holland Amsterdam · Zbl 0369.94008  Maneri, Carl; Silverman, Robert, A vector-space packing problem, J. algebra, 4, 321-330, (1966) · Zbl 0151.01602  Rédei, L., Lacunary polynomials over finite fields, (1973), North-Holland Amsterdam, Translated from the German by I. Földes · Zbl 0255.12009  Segre, Beniamino, Curve razionali normali e k-archi negli spazi finiti, Ann. mat. pura appl. (4), 39, 357-379, (1955) · Zbl 0066.14001  Segre, Beniamino, Introduction to Galois geometries, Atti accad. naz. lincei mem. cl. sci. fis. mat. natur. sez. I (8), 8, 133-236, (1967) · Zbl 0194.21503  Silverman, Robert, A metrization for power-sets with applications to combinatorial analysis, Canad. J. math., 12, 158-176, (1960) · Zbl 0092.01201  Szőnyi, T., k-sets in PG(2,q) having a large set of internal nuclei, (), 449-458 · Zbl 0945.51521  Szőnyi, Tamás, Around Rédei’s theorem, Discrete math., 208/209, 557-575, (1999), Combinatorics (Assisi, 1996) · Zbl 0952.11027  Thas, J.A., Normal rational curves and $$(q + 2)$$-arcs in a Galois space $$S_{q - 2, q}$$$$(q = 2^h)$$, Atti accad. naz. lincei rend. cl. sci. fis. mat. natur. (8), 47, 249-252, (1970), 1969 · Zbl 0189.51901  Thas, J.A., Complete arcs and algebraic curves in $$\mathit{PG}(2, q)$$, J. algebra, 106, 2, 451-464, (1987) · Zbl 0608.14016  J.A. Thas, Finite geometries, varieties and codes, in: Proceedings of the International Congress of Mathematicians, Extra vol. III, Berlin, 1998, 1998, pp. 397-408 (electronic) · Zbl 0905.51005  Wettl, F., Internal nuclei of k-sets in finite projective spaces of three dimensions, (), 407-419 · Zbl 0734.51008  Wicker, Stephen B., Error control systems for digital communication and storage, (1995), Prentice Hall Upper Saddle River, NJ · Zbl 0847.94001  Avi Wigderson, On the work of Madhu Sudan, Notices Amer. Math. Soc., 2002, pp. 45-50 · Zbl 1159.01352
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