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j aj00 : q-ary coef cient vector of j0 -th power of a0 , where a0 is a root of 2nd degree j q-ary primitive polynomial, 0 j0 q, j aj11 : q-ary coef cient vector of j1 -th power of a1 , where a1 is a root of 2l-th degree j q-ary primitive polynomial, 0 j1 n1 2. (d) Show that the following H matrix is an example of a (64, 57) B3=12 ECjSEC code over GF 3 with 7 check symbols, where a0 is a root of 2nd-degree 3-ary primitive polynomial g0 x x2 2x 2, and a1 is a root of 6th-degree 3-ary primitive polynomial g1 x x6 2x 2:
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001001000001 002001001000 010010000010 020010010000 100100000100 200100100000 000000000000
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1220122021020022112111011010010012100011000010000010 0121012201220210200221121110110100100121000110000100 1210122012202102002211211101101001001210001100001000 2101220122021020022112111011010010012100011000010000 1012201220210200221121110110100100121000110000100000 0122012202102002211211101101001001210001100001000000 1111111111111111111111111111111111111111111111111111
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[ASHL00] J. Ashley, M. P. Bermal, H. Coufal, H. Guenther, et al., Holographic Data Storage, IBM J. Res. Dev., 44 (May 2000): 341 366. [BELL90] T. C. Bell, et al., Text Compression, Prentice-Hall (1990). [BETZ98] G. A. Betzos, M. S. Porter, J. F. Hutton, and P. A. Mitkas, Optical Storage Interactive Simulator (OASIS): An Interactive Tool for the Analysis of Page-Oriented Optical Memories, Appl. Optics, 37 (September 1998): 6115 6126. [BURR97] G. W. Burr and B. Marcus, Coding Tradeoffs for High-Density Holographic Data Storage, Proc. SPIE, 3802 (July 1999): 18 29. [CHEN01] H. Chen, M. Kitakami, and E. Fujiwara, Burst Error Recovery for VF Arithmetic Coding, IEICE Trans. Fundamentals, E84-A (April 2001): 1050 1063. [CHO98] S. H. Cho, R. Kohno, and J. H. Park, Non-Proper Variable-to-Fixed Length Arithmetic Coding, IEICE Trans. Fundamentals, E81-A (August 1998): 1739 1747. [CHOU98] W. Chou and M. A. Neifeld, Interleaving and Error Correction in Volume Holographic Memory Systems, Appl. Optics, 37 (October 1998): 6951 6968. [FUJI95] E. Fujiwara and M. Kitakami, A Class of Optimal Fixed-Byte Error Protection Codes for Computer Systems, Dig. 25th IEEE Int. Symp. on Fault-Tolerant Computing (June 1995): 310 319. [FUJI98] E. Fujiwara, T. Ritthongpitak, and M. Kitakami, Optimal Two-Level Unequal Error Control Codes for Computer Systems, IEEE Trans. Comput., 47 (December 1998): 1313 1325. [FUJI02] E. Fujiwara, K. Namba and M. Kitakami, Parallel Decoding for Burst Error Control Codes (in Japanese), IEICE Trans. Fundamentals, J85-A (November 2002): 1284 1295. (Translated into English in Electron. Commun. Japan, P. 3, 87 [January 2004]: 38 48.)
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CODES FOR UNEQUAL ERROR CONTROL / PROTECTION (UEC / UEP)
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[FUJI03] E. Fujiwara and M. Kitakami, Unequal Error Protection in Ziv-Lempel Coding, IEICE Trans. Info. Syst., E86-D (December 2003): 2595 2600. [GUPT75] S. N. Gupta, On UEP Burst Codes, J. Math. Sci., 10, (1975): 21 27. [HAYA00] T. Hayashi and E. Fujiwara, Bit and Byte Error Protection Codes with Two Protection Levels (in Japanese), IEICE Trans. Fundamentals, J83-A (February 2000): 196 207. [IWAS97] A. Iwasaki and E. Fujiwara, SEC-DED Codes with Byte Error Protection Capabilities, (in Japanese), Proc. IEICE Fundamental Convention, A-6-4 (1997): 122. [KANE03] H. Kaneko and E. Fujiwara, Optimal Two-Level q-Ary Unequal Error Control Codes, Proc. 2003 IEEE Int. Symp. on Information Theory (June 2003): 215. [KING00] B. M. King and M. A. Neifeld, Sparse modulation coding for increased capacity in volume holographic storage, Appl. Optics, 39 (December 2000): 6681 6688. [LIN83] S. Lin and D. J. Costello Jr., Error Control Coding: Fundamentals and Applications, Prentice-Hall (1983). [LO96] J. C. Lo, M. Kitakami, and E. Fujiwara, Reliable Logic Circuits with Byte Error Control Codes A Feasibility Study, Proc. IEEE Int. Symp. on Defect and Fault Tolerance in VLSI Systems (November 1996): 286 295. [MASN67] B. Masnick and J. Wolf, On Linear Unequal Error Protection Codes, IEEE Trans. Info. Theory, IT-3 (October 1967): 600 607. [MORE94] R. H. Morelos-Zaragoza and S. Lin, On a Class of Optimal Nonbinary Linear UnequalError-Protection Codes for Two Sets of Messages, IEEE Trans. Info. Theory, IT-40 (January 1994): 196 200. [NAMB02] K. Namba and E. Fujiwara, Two-Level Unequal Error Protection Codes with Burst and Bit Error Correcting Capabilities, IEICE Trans. Fundamentals, E-85-A (June 2002): 1426 1430. [NAMB03] K. Namba, Unequal Error Control Codes with Two-Level Burst and Bit Error Correcting Capabilities (in Japanese), IEICE Trans. Fundamentals, J-86-A, (May 2003): 578 586. This is translated in English in Electronics and Communications in Japan Part II : Fundamental Electronics Science, 87 (April 2004) : 21 29. [PETE72] W. W. Peterson and E. J. Weldon Jr., Error-Correcting Codes, 2d ed., MIT Press (1972). [REIG60] S. H. Reiger, Codes for the Correction of Clustered Errors, IRE Trans. Info. Theory, IT-6 (March 1960): 16 21. [RITT95] T. Ritthongpitak and E. Fujiwara, A Class of Optimal Fixed-Byte Error Protection Codes, Proc. 1995 IEEE Int. Symp. on Information Theory (September 1995): 148. [RITT96] T. Ritthongpitak, M. Kitakami, and E. Fujiwara, Optimal Two-Level Unequal Error Control Codes for Computer Systems, Dig. 26th IEEE Int. Symp. on Fault-Tolerant Computing (June 1996): 190 199. [TILB89] H. V. Tilborg and M. Blaum, On Error-Correcting Balanced Codes, IEEE Trans. Info. Theory, 35 (September 1989): 1091 1095. [WELC84] T. A. Welch, ATechnique for High-Performance Data Compression, IEEE Compu., 17 (June 1984): 8 16. [ZIV77] J. Ziv and A. Lempel, A Universal Algorithm for Sequential Data Compression, IEEE Trans. Info. Theory, IT-23 (May 1977): 337 343. [ZIV78] J. Ziv and A. Lempel, Compression of Individual Sequences via Variable-Rate Coding, IEEE Trans. Info. Theory, IT-24(5) (September 1978): 530 536.
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CONTENTS 11.1 Tape Memory Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Optimal Rectangular Codes (ORC) . . . . . . . . . . . . . . . . . . 11.1.2 Adaptive Cross-Parity (AXP) Codes . . . . . . . . . . . . . . . . . . 11.1.3 Interleaved RS SbEC Codes for Mass Storage System (MSS) 11.2 Magnetic Disk Memory Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Fire Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Interleaved RS SbEC-DbED Codes . . . . . . . . . . . . . . . . . . 11.2.3 Computer-Generated Polynomial Codes . . . . . . . . . . . . . . . 11.2.4 Introduction to Disk Array Codes . . . . . . . . . . . . . . . . . . . 11.3 Optical Disk Memory Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Cross-interleaved RS Code (CIRC) . . . . . . . . . . . . . . . . . . 11.3.2 Long-Distance Code (LDC) . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 RS Product Codes for DVDs . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 466 475 481 487 488 492 497 497 500 500 505 507 509 512
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