Error-Correction of Multidimensional Bursts

A construction for D-dimensional binary codes of size n1timesn2timeshelliptimesnD correcting a single D-dimensional box error is presented. If the size of the box error is b1timesb2timeshelliptimesbD, bi odd, 1les i les D, and B = Pii D=1bi, then the redundancy of the code is at most [log2(n1n2hellip nD)] +B + (D-2)[log2 B] + [log2b1]. For a two-dimensional binary array of size n times n we present a code correcting an error whose shape is a Lee sphere with radius R. The redundancy of the code is at most [log2n2] + 2R2 + 2R + [2log2 (2R+1)]+1. This is also the redundancy of a binary code which corrects an arbitrary two-dimensional cluster-error of size 2R+1. A generalization for D-dimensional code which corrects either D-dimensional error whose shape is a Lee sphere or an arbitrary cluster-error is also given.

[1]  S. Golomb,et al.  Perfect Codes in the Lee Metric and the Packing of Polyominoes , 1970 .

[2]  L Hesselink,et al.  Volume Holographic Storage and Retrieval of Digital Data , 1994, Science.

[3]  Martin Bossert,et al.  Array Codes Correcting a Two-Dimensional Cluster of Errors , 1998, IEEE Trans. Inf. Theory.

[4]  Khaled Ahmed Sabry Abdel-Ghaffar.  An information- and coding-theoretic study of bursty channels with applications to computer memories (two-dimensional) , 1986 .

[5]  Claire Gu,et al.  Crosstalk limited storage capacity of volume holographic memory , 1992, Optical Society of America Annual Meeting.

[6]  Khaled A. S. Abdel-Ghaffar,et al.  On the existence of optimum cyclic burst-correcting codes , 1986, IEEE Trans. Inf. Theory.

[7]  Ron M. Roth,et al.  Reduced-Redundancy Product Codes for Burst Error Correction , 1998, IEEE Trans. Inf. Theory.

[8]  I. M. Boyarinov.  Two-dimensional array codes correcting rectangular burst errors , 2006, Probl. Inf. Transm..

[9]  Khaled A. S. Abdel-Ghaffar.  On the existence of optimum cyclic burst correcting codes over GF(q) , 1988, IEEE Trans. Inf. Theory.

[10]  Alexander Vardy,et al.  Two-dimensional interleaving schemes with repetitions: Constructions and bounds , 2002, IEEE Trans. Inf. Theory.

[11]  M A Neifeld,et al.  Error correction for increasing the usable capacity of photorefractive memories. , 1994, Optics letters.

[12]  S. H. Reiger,et al.  Codes for the correction of 'clustered' errors , 1960, IRE Trans. Inf. Theory.

[13]  HIDEKI IMAI,et al.  Two-dimensional Fire codes , 1973, IEEE Trans. Inf. Theory.

[14]  Moshe Schwartz,et al.  Two-dimensional cluster-correcting codes , 2005, IEEE Transactions on Information Theory.

[15]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[16]  Alexander Vardy,et al.  Interleaving Schemes for Multidimensional Cluster Errors , 1998, IEEE Trans. Inf. Theory.

[17]  Khaled A. S. Abdel-Ghaffar,et al.  Two-dimensional burst identification codes and their use in burst correction , 1988, IEEE Trans. Inf. Theory.

[18]  Steven A. Lis,et al.  Ultra-Dense Optical Mass Storage , 1991 .

[19]  Ron M. Roth,et al.  Introduction to Coding Theory , 2019, Discrete Mathematics.

[20]  D. Klarner,et al.  A Procedure for Improving the Upper Bound for the Number of n-Ominoes , 1972, Canadian Journal of Mathematics - Journal Canadien de Mathematiques.

[21]  Martin Bossert,et al.  Singleton-type bounds for blot-correcting codes , 1996, IEEE Trans. Inf. Theory.