Designs, Codes and Cryptography

, Volume 76, Issue 1, pp 49–79

# An improvement of the Feng–Rao bound for primary codes

• Olav Geil
• Stefano Martin
Article

## 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.

## Keywords

Affine variety code $$C_{ab}$$ curve Feng–Rao bound Generalised Hamming weight Minimum distance One-way well-behaving pair

## Mathematics Subject Classification

94B65 94B27 94B05

## Notes

### Acknowledgments

Part of this research was done while the second listed author was visiting East China Normal University. We are grateful to Professor Hao Chen for his hospitality. The authors also gratefully acknowledge the support from the Danish National Research Foundation and the National Science Foundation of China (Grant No. 11061130539) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography. The authors would like to thank Diego Ruano, Peter Beelen and Ryutaroh Matsumoto for pleasant discussions. Finally we are grateful to the anonymous reviewers for their careful reading and useful suggestions.

## References

1. 1.
Andersen H.E., Geil O.: Evaluation codes from order domain theory. Finite Fields Appl. 14(1), 92–123 (2008).Google Scholar
2. 2.
Borges H., Conceição R.: On the characterization of minimal value set polynomials. J. Number Theory 133(6), 2021–2035 (2013).Google Scholar
3. 3.
Cox D.A., Little J., O’Shea D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, Heidelberg (1997).Google Scholar
4. 4.
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).Google Scholar
5. 5.
Fitzgerald J., Lax R.F.: Decoding affine variety codes using Gröbner bases. Des. Codes Cryptogr. 13(2), 147–158 (1998).Google Scholar
6. 6.
Geil O.: On codes from norm-trace curves. Finite Fields Appl. 9(3), 351–371 (2003).Google Scholar
7. 7.
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).Google Scholar
8. 8.
Geil O., Høholdt T.: Footprints or generalized Bezout’s theorem. IEEE Trans. Inform. Theory 46(2), 635–641 (2000).Google Scholar
9. 9.
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. 10.
Geil O., Pellikaan R.: On the structure of order domains. Finite Fields Appl. 8(3), 369–396 (2002).Google Scholar
11. 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).Google Scholar
12. 12.
Geil O., Matsumoto R., Ruano D.: Feng–Rao decoding of primary codes. Finite Fields Appl. 23, 35–52 (2013).Google Scholar
13. 13.
Grassl M.: Code tables: bounds on the parameters of various types of codes. http://www.codetables.de (2013).
14. 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).Google Scholar
15. 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).Google Scholar
16. 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).Google Scholar
17. 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).Google Scholar
18. 18.
Matsumoto R.: The $${C}_{ab}$$ curve. http://www.rmatsumoto.org/cab (1998).
19. 19.
Miura S.: Algebraic geometric codes on certain plane curves. Electron. Commun. Jpn. 76(12):1–13 (1993). (in Japanese).Google Scholar
20. 20.
Miura S.: Study of error-correcting codes based on algebraic geometry. PhD thesis, University of Tokyo (1997). (in Japanese).Google Scholar
21. 21.
Miura S.: Linear codes on affine algebraic curves. Trans. IEICE J81–A(10), 1398–1421 (1998). (in Japanese).Google Scholar
22. 22.
Pellikaan R.: On the existence of order functions. J. Stat. Plan. Inference 94(2), 287–301 (2001).Google Scholar
23. 23.
Rédei L.: Lacunary Polynomials Over Finite Fields. North-Holland Publ. Comp, Amsterdam (1973).Google Scholar
24. 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).Google Scholar
25. 25.
Wei V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inform. Theory 37(5), 1412–1418 (1991).Google Scholar