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Designs, Codes and Cryptography

Erschienen in:

01.07.2015

An improvement of the Feng–Rao bound for primary codes

verfasst von: Olav Geil, Stefano Martin

Erschienen in: Designs, Codes and Cryptography | Ausgabe 1/2015

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Abstract

We present a new bound for the minimum distance of a general primary linear code. For affine variety codes defined from generalised \(C_{ab}\) polynomials the new bound often improves dramatically on the Feng–Rao bound for primary codes (Andersen and Geil, Finite Fields Appl 14(1):92–123, 2008; Geil et al., Lecture Notes in Computer Science 3857: 295–306, 2006). The method does not only work for the minimum distance but can be applied to any generalised Hamming weight.

Vorheriger Artikel Optimal codes as Tanner codes with cyclic component codes

Nächster Artikel The second generalized Hamming weight of certain Castle codes

Andersen H.E., Geil O.: Evaluation codes from order domain theory. Finite Fields Appl. 14(1), 92–123 (2008).

Borges H., Conceição R.: On the characterization of minimal value set polynomials. J. Number Theory 133(6), 2021–2035 (2013).

Cox D.A., Little J., O’Shea D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, Heidelberg (1997).

Feng G.L., Rao T.R.N.: A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inform. Theory 40(4), 1003–1012 (1994).

Fitzgerald J., Lax R.F.: Decoding affine variety codes using Gröbner bases. Des. Codes Cryptogr. 13(2), 147–158 (1998).

Geil O.: On codes from norm-trace curves. Finite Fields Appl. 9(3), 351–371 (2003).

Geil O.: Evaluation codes from an affine variety code perspective. In: Martínez-Moro E., Munuera C., Ruano D. (eds.) Advances in Algebraic Geometry Codes. Coding Theory and Cryptology, vol. 5, pp. 153–180. World Scientific, Singapore (2008).

Geil O., Høholdt T.: Footprints or generalized Bezout’s theorem. IEEE Trans. Inform. Theory 46(2), 635–641 (2000).

Geil O., Martin S.: Further improvements on the Feng–Rao bound for dual codes. Finite Fields Appl. arXiv, preprint arXiv:1305.1091 (2013) (to appear).

10.

Geil O., Pellikaan R.: On the structure of order domains. Finite Fields Appl. 8(3), 369–396 (2002).

11.

Geil O., Thommesen C.: On the Feng–Rao bound for generalized Hamming weights. In: Fossorier M.P.C., Imai H., Lin S., Poli A. (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science, vol. 3857, pp. 295–306. Springer, Heidelberg (2006).

12.

Geil O., Matsumoto R., Ruano D.: Feng–Rao decoding of primary codes. Finite Fields Appl. 23, 35–52 (2013).

13.

Grassl M.: Code tables: bounds on the parameters of various types of codes. http://www.codetables.de (2013).

14.

Hernando F., Marshall K., O’Sullivan M.E.: The dimension of subcode-subfields of shortened generalized Reed–Solomon codes. Des. Codes Cryptogr. 16, 131–142 (2013).

15.

Høholdt T., van Lint J.H., Pellikaan R.: Algebraic geometry codes. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, pp. 871–961. Elsevier, Amsterdam (1998).

16.

Kurihara J., Uyematsu T., Matsumoto R.: Secret sharing schemes based on linear codes can be precisely characterized by the relative generalized Hamming weight. IEICE Trans. Fundam. E95–A(11), 2067–2075 (2012).

17.

Luo Y., Mitrpant C., Vinck A.J.H., Chen K.: Some new characters on the wire-tap channel of type II. IEEE Trans. Inform. Theory 51(3), 1222–1229 (2005).

18.

Matsumoto R.: The \({C}_{ab}\) curve. http://www.rmatsumoto.org/cab (1998).

19.

Miura S.: Algebraic geometric codes on certain plane curves. Electron. Commun. Jpn. 76(12):1–13 (1993). (in Japanese).

20.

Miura S.: Study of error-correcting codes based on algebraic geometry. PhD thesis, University of Tokyo (1997). (in Japanese).

21.

Miura S.: Linear codes on affine algebraic curves. Trans. IEICE J81–A(10), 1398–1421 (1998). (in Japanese).

22.

Pellikaan R.: On the existence of order functions. J. Stat. Plan. Inference 94(2), 287–301 (2001).

23.

Rédei L.: Lacunary Polynomials Over Finite Fields. North-Holland Publ. Comp, Amsterdam (1973).

24.

Salazar G., Dunn D., Graham S.B.: An improvement of the Feng–Rao bound on minimum distance. Finite Fields Appl. 12, 313–335 (2006).

25.

Wei V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inform. Theory 37(5), 1412–1418 (1991).

Titel: An improvement of the Feng–Rao bound for primary codes
verfasst von: Olav Geil
Stefano Martin
Publikationsdatum: 01.07.2015
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 1/2015
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-014-9983-z

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An improvement of the Feng–Rao bound for primary codes

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