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Designs, Codes and Cryptography

Erschienen in:

23.10.2017

On critical exponents of Dowling matroids

verfasst von: Yoshitaka Koga, Tatsuya Maruta, Keisuke Shiromoto

Erschienen in: Designs, Codes and Cryptography | Ausgabe 9/2018

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Abstract

The critical problem in matroid theory is the problem to determine the critical exponent of a given representable matroid over a finite field. In this paper, we study the critical exponents of a class of representable matroids over finite fields, called Dowling matroids. Then the critical problem for a Dowling matroid is corresponding to the classical problem in coding theory to determine the maximum dimension k such that there exists an \([n,k,d]_q\) code for given n, d and q. We give a necessary and sufficient condition on the critical exponents of Dowling matroids by using a coding theoretical approach.

Vorheriger Artikel Zero-correlation attacks: statistical models independent of the number of approximations

Nächster Artikel Flag-transitive point-quasiprimitive 2- designs

Abbe E., Alon N., Bandeira A.S.: Linear Boolean classification, coding and “the critical problem”. In: 2014 IEEE International Symposium on Information Theory (ISIT), pp. 1231–1235 (2014).

Asano T., Nishizeki T., Oxley J., Saito N.: A note on the critical problem for matroids. Eur. J. Comb. 5, 93–97 (1984).MathSciNetCrossRefMATH

Ball S.: Table of bounds on three dimensional linear codes or \((n,r)\) arcs in PG\((2,q)\). https://mat-web.upc.edu/people/simeon.michael.ball/codebounds.html/. Accessed 19 Jan 2017

Britz T.: Extensions of the critical theorem. Discret. Math. 305, 55–73 (2005).MathSciNetCrossRefMATH

Britz T., Shiromoto K.: On the covering dimension of a linear code. IEEE Trans. Inf. Theory 62, 2694–2701 (2016).MathSciNetCrossRefMATH

Crapo H.H., Rota G.-C.: On the Foundations of Combinatorial Theory: Combinatorial Geometries, Preliminary edn. MIT, Cambridge (1970).MATH

Dowling, T.A.: Codes, packings and the critical problem. In: Applicazioni (Univ. Perugia, Perugia, 1970), Ist. Mat., Univ. Perugia, Perugia, pp. 209–224 (1971).

Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de/. Accessed 19 Jan 2017.

Hadwiger H.: Über eine Klassifikation der Streckenkomplexe. Vierteljschr. Naturforsch. Ges. Zürich 88, 133–143 (1943).MathSciNetMATH

10.

Hamada N., Tamari F.: Construction of optimal codes and optimal fractional factorial designs using linear programming. Ann. Discret. Math. 6, 175–188 (1980).MathSciNetCrossRefMATH

11.

Hill R.: Optimal Linear Codes, Cryptography and Coding, II (Cirencester, 1989). Institute of Mathematics and Its Applications Conference Series, vol. 33, pp. 75–104. Oxford University Press, New York (1992).

12.

Hirschfeld J.W.P., Storme L.: The packing problem in statistics, coding theory and finite projective spaces. In: Proceedings of the Bose Memorial Conference (Colorado, June 7–11). J. Stat. Plan. Inference 72, pp. 355–380 (1998) (Updated version) (1995).

13.

Huffman W., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).CrossRefMATH

14.

Johnsen T., Shiromoto K., Verdure H.: A generalization of Kung’s theorem. Des. Codes Cryptogr. 81, 169–178 (2016).MathSciNetCrossRefMATH

15.

Kung J.P.S.: Critical problems. In: Matroid Theory (Seattle, WA, 1995). Contemporary Mathematics, vol. 197. American Mathematical Society, Providence, pp. 1–127 (1996).

16.

Kung J.P.S.: Critical exponents, colines, and projective geometries. Comb. Probab. Comput. 9, 355–362 (2000).MathSciNetCrossRefMATH

17.

Lindström B.: On the chromatic number of regular matroids. J. Comb. Theory Ser. B 24, 367–369 (1978).MathSciNetCrossRefMATH

18.

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

19.

Maruta T.: On the achievement of the Griesmer bound. Des. Codes Cryptogr. 12, 83–87 (1997).MathSciNetCrossRefMATH

20.

Maruta T.: Construction of optimal linear codes by geometric puncturing. Serdica J. Comput. 7, 73–80 (2013).MathSciNetMATH

21.

Maruta T.: Griesmer bound for linear codes over finite fields. http://www.mi.s.osakafu-u.ac.jp/~maruta/griesmer/. Accessed 19 Jan 2017.

22.

Oxley J.: Colouring, packing and the critical problem. Q. J. Math. Oxf. 29(2), 11–22 (1978).MathSciNetCrossRefMATH

23.

Oxley J.: Matroid Theory. Oxford University Press, New York (1992).MATH

24.

Tutte W.T.: A contribution on the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954).MathSciNetCrossRefMATH

25.

Welsh D.J.A.: Matroid Theory. Academic Press, London (1976).MATH

Titel: On critical exponents of Dowling matroids
verfasst von: Yoshitaka Koga
Tatsuya Maruta
Keisuke Shiromoto
Publikationsdatum: 23.10.2017
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 9/2018
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-017-0431-8

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