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The Art of Error Correcting Coding phần 4

để sửa chữa lỗi với single-bit lên đến 8 bit của chi phí, tối đa là 72 bit (64 bit dữ liệu cộng với 8 bit dư thừa) trong tổng số. Đương nhiên, vì 72 là không hình thức 2m - 1, không có mã số theo chu kỳ nghiên cứu cho đến nay có thể được áp dụng trực tiếp. Một giải pháp có thể là sử dụng một chu kỳ | BINARY CYCLIC CODES AND BCH CODES 71 17. Let C be a double-error-correcting 15 7 5 BCH code. Suppose that the all-zero code word is transmitted and that there are erasures in the last four consecutive positions so that r x f fx fx2 I x where f denotes an erasure. Find the most likely code polynomial that was transmitted. 18. Prove the existence of a binary cyclic 63 58 code. Hint How many binary cyclic codes of length 63 are there 19. Let Cl be a double-error-correcting BCH 15 7 5 code and C2 be a triple-errorcorrecting BCH 15 5 7 code. a Find generator matrices G and G2 of C1 and C2 respectively. b Compute the weight distributions w C1 and w C2 . Hint You may use a computer program for this task. c Estimate the performance of codes C1 and C2 BER curves with soft-decision decoding over an AWGN channel. Compare the curves with respect to uncoded BPSK modulation. What are the respective coding gains 20. Let At be an í X Í Vandermonde matrix over a field. a Show that det At 0 that is the rank of At is . b Show that any V X V submatrix AV of At has rank V. 21. Let 2 1 n1 n2 where n1 1 and n2 1. a Show that the element p an1 where a is a primitive element in GF 2m has order n2. That is n2 is the smallest positive integer such that pn- 1. b Using p a binary nonprimitive BCH code can be constructed. With n 63 determine the generator polynomial of a binary nonprimitive BCH 21 12 5 code. 4 Nonbinary BCH codes Reed-Solomon codes In this chapter one of the most celebrated class of ECC schemes is introduced and their encoding and decoding algorithms explained. Reed-Solomon RS codes have found numerous applications in digital storage and communication systems. Examples include the famous RS 255 223 33 code for NASA space communications shortened RS codes over GF 28 for CD-ROM DVD and Terrestrial Digital HDTV transmission applications an extended RS 128 122 7 code over GF 27 for cable modems among many others. 4.1 RS codes as polynomial codes Similar to Reed-Muller RM codes RS .