Abstract
A′(n, d, e), the smallest ℓ for which every binary error-correcting code of length n and minimum distance d is decodable with a list of size ℓ up to radius e, is determined for all d ≥ 2e − 3. As a result, A′(n, d, e) is determined for all e ≤ 4, except for 42 values of n.
Similar content being viewed by others
References
Ahlswede R.: Channel capacities for list codes. J. Appl. Probab. 10(4), 824–836 (1973)
Bassalygo L.A.: New upper bounds for error-correcting codes. Problemy Peredači Informacii 1(vyp. 4), 41–44 (1965)
Blinovsky V.: Bounds for codes in the case of list decoding of finite volume. Probl. Inform. Transm. 22(1), 7–19 (1986)
Blinovsky V.: Asymptotic Combinatorial Coding Theory. The Kluwer International Series in Engineering and Computer Science, 415. Kluwer Academic Publishers, Boston (1997).
Brouwer A.E.: On the packing of quadruples without common triples. Ars Combin. 5, 3–6 (1978)
Brouwer A.E.: Optimal packings of K 4’s into a K n . J. Comb. Theory A 26(3), 278–297 (1979)
Brouwer A.E., Shearer J.B., Sloane N.J.A., Smith W.D.: A new table of constant weight codes. IEEE Trans. Inform. Theory 36(6), 1334–1380 (1990)
Cassuto Y., Bruck J.: A combinatorial bound on the list size. Technical Report, California Institute of Technology, Pasedna, CA (2004).
Elias P.: List decoding for noisy channels. Technical report 335. Research Laboratory of Electronics, Massachusetts Institute of Technology (1957).
Elias P.: Error-correcting codes for list decoding. IEEE Trans. Inform. Theory 37(1), 5–12 (1991)
Forney G.D.: Exponential error bounds for erasure, list and decision feedback schemes. IEEE Trans. Inform. Theory 14, 549–557 (1968)
Goldreich O., Levin L.A.: A hard-core predicate for all one-way functions. In: STOC 1989: Proceedings of the 21st Annual ACM Symposium on Theory of Computing, 15–17 May 1989, Seattle, Washington, USA, pp. 25–32. ACM Press (1989).
Guruswami V.: List Decoding of Error-Correcting Codes, Lecture Notes in Computer Science, vol. 3282. Springer-Verlag, Berlin (2004).
Guruswami V., Sudan M.: Extensions to the Johnson bound. Unpublished manuscript (2001).
Hanani H.: On quadruple systems. Can. J. Math. 12, 145–157 (1960)
Ji L.: Asymptotic determination of the last packing number of quadruples. Des. Codes Cryptogr. 38(1), 83–95 (2006)
Johnson S.M.: Upper bounds for constant weight error-correcting codes. Discrete Math. 3, 109–124 (1972)
Schönheim J.: On maximal systems of k-tuples. Studia Sci. Math. Hungar 1, 363–368 (1966)
Shannon C.E., Gallager R.G., Berlekamp E.R.: Lower bounds to error probability for coding on discrete memoryless channels. I. Inform. Control 10, 65–103 (1967)
Shannon C.E., Gallager R.G., Berlekamp E.R.: Lower bounds to error probability for coding on discrete memoryless channels. II. Inform. Control 10, 522–552 (1967)
Spencer J.: Maximal consistent families of triples. J. Comb. Theory 5, 1–8 (1968)
Sudan M.: Decoding of Reed Solomon codes beyond the error-correction bound. J. Complex. 13(1), 180–193 (1997)
Wozencraft J.M.: List decoding. Research Laboratory of Electronics, Massachusetts Institute of Technology, Progress report 48, pp. 90–95 (1958).
Zyablov V.V., Pinsker M.S.: List cascade decoding. Probl. Inform. Transm. 17(4), 236–240 (1982)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. A. Zinoviev.
Rights and permissions
About this article
Cite this article
Chee, Y.M., Ge, G., Ji, L. et al. List decodability at small radii. Des. Codes Cryptogr. 61, 151–166 (2011). https://doi.org/10.1007/s10623-010-9445-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-010-9445-1