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Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 3/2022

25-01-2022

From primary to dual affine variety codes over the Klein quartic

Author: Olav Geil

Published in: Designs, Codes and Cryptography | Issue 3/2022

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Abstract

In Geil and Özbudak (Cryptogr Commun 11(2):237–257, 2019) a novel method was established to estimate the minimum distance of primary affine variety codes and a thorough treatment of the Klein quartic led to the discovery of a family of primary codes with good parameters, the duals of which were originally treated in Kolluru et al (Appl Algebra Eng Commun Comput 10(6):433-464, 2000)[Ex. 3.2, Ex. 4.1]. In the present work we translate the method from Geil and Özbudak (Cryptogr Commun 11(2):237–257, 2019) into a method for also dealing with dual codes and we demonstrate that for the considered family of dual affine variety codes from the Klein quartic our method produces much more accurate information than what was found in Kolluru et al (Appl Algebra Eng Commun Comput 10(6):433-464, 2000). Combining then our knowledge on both primary and dual codes we determine asymmetric quantum codes with desirable parameters.
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Footnotes
1
We remark that in cases where for a given index i there exists an \(s \in {\mathbb {F}}_q\) such that for no \(P_j\) the ith coordinate is s we may leave out this value in the ith product of (6) simplifying the calculations.
 
Literature
1.
Aly S.A., Klappenecker A., Sarvepalli P.K.: Remarkable degenerate quantum stabilizer codes derived from duadic codes. In: 2006 IEEE International Symposium on Information Theory, pp. 1105–1108 (2006).
2.
3.
Beelen P., Datta M.: Generalized Hamming weights of affine Cartesian codes. Finite Fields Appl. 51, 130–145 (2018).MathSciNetCrossRef
4.
Carvalho C., Chara M., Quoos L.: On evaluation codes coming from a tower of function fields. J. Symbolic Comput. 89, 121–128 (2018).MathSciNetCrossRef
5.
Cox D.A., Little J., O’Shea D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, vol. 10. Springer, New York (1997).CrossRef
6.
Duursma I.M., Kirov R.: An extension of the order bound for AG codes. In: Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes. Lecture Notes in Comput. Sci., vol. 5527, pp. 11–22. Springer, Berlin (2009).
7.
Duursma I.M., Kirov R., Park S.: Distance bounds for algebraic geometric codes. J. Pure Appl. Algebra 215(8), 1863–1878 (2011).MathSciNetCrossRef
8.
9.
10.
Ezerman M.F., Jitman S., Solé P.: Xing-Ling codes, duals of their subcodes, and good asymmetric quantum codes. Des. Codes Cryptogr. 75(1), 21–42 (2015).MathSciNetCrossRef
11.
Feng G.L., Rao T.R.N.: Decoding algebraic-geometric codes up to the designed minimum distance. IEEE Trans. Inf. Theory 39(1), 37–45 (1993).MathSciNetCrossRef
12.
Feng G.L., Rao T.R.N.: A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inf. Theory 40(4), 1003–1012 (1994).MathSciNetCrossRef
13.
Feng G.L., Rao T.R.N.: Improved geometric Goppa codes part I: basic theory. IEEE Trans. Inf. Theory 41(6), 1678–1693 (1995).CrossRef
14.
Fitzgerald J., Lax R.F.: Decoding affine variety codes using Gröbner bases. Des. Codes Cryptogr. 13(2), 147–158 (1998).MathSciNetCrossRef
15.
García-Marco I., Márquez-Corbella I., Ruano D.: High dimensional affine codes whose square has a designed minimum distance. Des. Codes Cryptogr. 88(8), 1653–1672 (2020).MathSciNetCrossRef
16.
Geil O.: Evaluation codes from an affine variety code perspective. In: Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol., vol. 5, pp. 153–180. World Sci. Publ., Hackensack (2008).
17.
Geil O., Martin S.: Further improvements on the Feng-Rao bound for dual codes. Finite Fields Appl. 30, 33–48 (2014).MathSciNetCrossRef
18.
Geil O., Martin S.: An improvement of the Feng-Rao bound for primary codes. Des. Codes Cryptogr. 76(1), 49–79 (2015).MathSciNetCrossRef
19.
20.
21.
22.
Grassl M., Lu S., Zeng B.: Codes for simultaneous transmission of quantum and classical information. In: 2017 IEEE International Symposium on Information Theory (ISIT), pp. 1718–1722 (2017).
23.
Høholdt T.: On (or in) Dick Blahut’s’ footprint’. Codes, Curves and Signals, pp. 3–9 (1998).
24.
Høholdt T., van Lint J.H., Pellikaan R.: Algebraic geometry codes. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, vol. 1, pp. 871–961. Elsevier, Amsterdam (1998).
25.
26.
Kolluru M.S., Feng G.L., Rao T.R.N.: Construction of improved geometric Goppa codes om Klein curves and Klein-like curves. Appl. Algebra Eng. Commun. Comput. 10(6), 433–464 (2000).CrossRef
27.
López H.H., Soprunov I., Villarreal R.H.: The dual of an evaluation code. Des. Codes Cryptogr. 89(7), 1367–1403 (2021).MathSciNetCrossRef
28.
Matsumoto R.: Two Gilbert-Varshamov-type existential bounds for asymmetric quantum error-correcting codes. Quantum Inf. Process. 16(12), 285 (2017).MathSciNetCrossRef
29.
Matsumoto R., Miura S.: On the Feng-Rao bound for the \({\cal{L}}\)-construction of algebraic geometry codes. IEICE Trans. Fundam. 5, 926–930 (2000) E83-A.
30.
Miura S.: Linear codes on affine algebraic curves. Trans. IEICE 10, 1398–1421 (1998) J81-A.
31.
Patanker N., Singh S.K.: Affine variety codes over a hyperelliptic curve. Probl. Inf. Transm. 57, 84–97 (2021).CrossRef
32.
Pellikaan R.: On the efficient decoding of algebraic-geometric codes. In: Eurocode ’92 (Udine, 1992), CISM Courses and Lect., vol. 339, pp. 231–253. Springer, Vienna (1993).
33.
Salazar G., Dunn D., Graham S.B.: An improvement of the Feng-Rao bound on minimum distance. Finite Fields Appl. 12, 313–335 (2006).MathSciNetCrossRef
34.
Sarvepalli P.K., Klappenecker A., Rötteler M.: Asymmetric quantum codes: constructions, bounds and performance. In: Proc. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci., vol. 465, pp. 1645–1672. The Royal Society (2009).
Metadata
Title
From primary to dual affine variety codes over the Klein quartic
Author
Olav Geil
Publication date
25-01-2022
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 3/2022
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-021-00990-5

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