Abstract
Three-dimensional cyclic array codes over F q that can correct single three-dimensional bursts (or clusters) of errors are considered. The class cyclic three-dimensional burst-error-correcting array codes, called three-dimensional Fire codes, is constructed. Several important properties such as the burst-error-correcting capability and the positions of the parity-check symbols are presented. Also, encoding and decoding algorithms are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blaum M., Bruck J., Vardy A.: Interleaving schemes for multidimensional cluster errors. IEEE Trans. Inf. Theory IT-44, 730–743 (1998)
Reiger S.H.: Codes for correction of clustered errors. IRE Trans. Inf. Theory IT-6, 16–21 (1960)
Peterson W.: Error-Correcting Codes. MIT Press, Cambridge (1961)
Abdel-Ghaffar K.A.S., McEliece R.J., van Tilborg H.C.A.: Two-dimensional burst identification codes and their use in burst correction. IEEE Trans. Inf. Theory 34, 494–504 (1988)
Boyarinov I.M.: Two-dimensional array codes correcting rectangular burst errors. Probl. Inf. Transm. 42(2), 90–105 (2006)
Etzion T., Yaakobi E.: Error-correction of multidimensional bursts. IEEE Trans. Inf. Theory IT-55, 961–976 (2009)
Boyarinov I.M.: Three-dimensional cluster-error-correcting array codes. In: 2002 International Symposium on Information Theory, Lausanne, p. 118 (2002).
Boyarinov I.M.: Three-dimensional array SCEC codes with small excess redundancy. In: Tenth International Workshop on Algebraic and Combinatoric Coding Theory, Zvenigorod (2006).
Blaum M., Farrell P.G.: Array codes for cluster-error correction. Electron. Lett. 30, 1752–1753 (1994)
Boyarinov I.M.: Three-dimensional array codes correcting 2 × 2 × 2-clusters of errors. In: Eleventh International Workshop on Algebraic and Combinatoric Coding Theory, Pamporovo, pp. 46–50 (2008).
Boyarinov I.: Three-dimensional Fire codes. In: Seventh International Workshop on Coding and Cryptography (WCC 2011), Paris (2011).
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
Nomura T., Miyakawa H., Imai H., Fukuda A.: A theory of two-dimensional linear recurring arrays. IEEE Trans. Inf. Theory 18, 775–785 (1972)
Imai H.: Two-dimensional Fire codes. IEEE Trans. Inf. Theory 19, 786–806 (1973)
Ikai T., Kosako H., Kojima Y.: On two-dimensional cyclic codes. Trans. Inst. Electron. Commun. Eng. Jpn. 57A, 279–286 (1974)
Imai H.: A theory of two-dimensional cyclic codes. Inf. Control 34, 1–21 (1977)
Fire P.: A class of multiple-error-correcting binary codes for non-independent errors. Sylvania Report No. RSL-E-2, Sylvania Electronic Defense Laboratory, Reconnaissance Systems Division, Mountain View, March (1959).
Lin S., Costello D.J.: Error Control Coding: Fundamentals and Applications. Prentice-Hall, Englewood Cliffs (1983).
Cox D., Little J., O’Shea D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd edn. Springer, New York (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
Rights and permissions
About this article
Cite this article
Boyarinov, I.M. Three-dimensional cyclic Fire codes. Des. Codes Cryptogr. 66, 363–373 (2013). https://doi.org/10.1007/s10623-012-9717-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-012-9717-z