Abstract
For an arbitrary prime power q, let α q be the standard function in the asymptotic theory of codes, that is, α q (δ) is the largest asymptotic information rate that can be achieved by a sequence of q-ary codes with a given asymptotic relative minimum distance δ. A central problem in the asymptotic theory of codes is to find lower bounds on α q (δ). In recent years several authors established various lower bounds on α q (δ). In this paper, we present a further improved lower bound by extending a result of Niederreiter and Özbudak (Finite Fields Appl 13: 423–443, 2007). In particular, we show that the bound 1 − δ − A(q)−1 + log q (1 + 2/q 3) + log q (1 + (q − 1)/q 6) can be achieved for certain values of q and certain ranges of δ.
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References
Bezerra J., Garcia A., Stichtenoth H.: An explicit tower of function fields over cubic finite fields and Zink’s lower bound. J. Reine Angew. Math. 589, 159–199 (2005)
Elkies N.D.: Still better nonlinear codes from modular curves. http://arxiv.org/abs/math.NT/0308046 (2003).
Ihara Y.: Some remarks on the number of rational points of algebraic curves of finite fields. J. Fac. Sci. Univ. Tokyo 28, 721–724 (1981)
Manin Y.I.: What is the maximal number of points on a curve over \({\mathbb{F}_2}\) . J. Fac. Sci. Univ. Tokyo 28, 712–720 (1981)
Maharaj H.: A note on further improvements of the TVZ-bound. IEEE Trans. Inform. Theory 53, 1210–1214 (2007)
Niederreiter H., Özbudak F.: Constructive asymptotic codes with an improvement on the Tsfasman-Vlǎduţ–Zink and Xing bounds. In: Feng, K.Q., Niederreiter, H., Xing, C.P. (eds) Coding, Cryptography and Combinatorics. Progress in Computer Science and Applied Logic, vol. 23, pp. 259–275. Birkhäuser, Basel (2004)
Niederreiter H., Özbudak F.: Further improvements on asymptotic bounds for codes using distinguished divisors. Finite Fields Appl. 13, 423–443 (2007)
Niederreiter H., Özbudak F.: Improved asymptotic bounds for codes using distinguished divisors of global function fields. SIAM J. Discrete Math. 21, 865–899 (2007)
Niederreiter H., Xing C.P.: Rational Points on Curves over Finite Fields: Theory and Applications. Cambridge University Press, Cambridge (2001)
Stichtenoth H.: Algebraic Function Fields and Codes. Springer-Verlag, Berlin (1993)
Stichtenoth H., Xing C.P.: Excellent nonlinear codes from algebraic function fields. IEEE Trans. Inform. Theory 51, 4044–4046 (2005)
Tsfasman M.A., Vlǎduţ S.G., Zink T.: Modular curves, Shimura curves and Goppa codes better than Gilbert-Varshamov bound. Math. Nachr. 109, 21–28 (1982)
Tsfasman M.A., Vlǎduţ S.G.: Algebraic-Geometric Codes. Kluwer, Dordrecht, The Netherlands (1991)
Vlǎduţ S.G.: An exhaustion bound for algebraic-geometric modular codes. Probl. Inf. Transm. 23, 22–34 (1987)
Xing C.P.: Algebraic-geometry codes with asymptotic parameters better than the Gilbert-Varshamov and Tsfasman-Vlǎduţ–Zink bounds. IEEE Trans. Inform. Theory 47, 347–352 (2001)
Xing C.P.: Nonlinear codes from algebraic curves improving the Tsfasman-Vlǎduţ–Zink bound. IEEE Trans. Inform. Theory 49, 1653–1657 (2003)
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Communicated by Shuhong Gao.
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Yang, S., Qi, L. On improved asymptotic bounds for codes from global function fields. Des. Codes Cryptogr. 53, 33–43 (2009). https://doi.org/10.1007/s10623-009-9289-8
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DOI: https://doi.org/10.1007/s10623-009-9289-8
Keywords
- Asymptotic theory of codes
- Algebraic geometry codes
- Global function fields
- Gilbert–Varshamov (GV) bound
- Tsfasman–Vlǎduţ-Zink (TVZ) bound