Error-Control Block Codes for Communications Engineers
An engineer s guide to the theory behind the common random-error-control block codes, and their applications in digital communications.
TABLE OF CONTENTS Error-Control Block Codes for Communications Engineers
3.5.4: Reed-Muller Codes
3.5.4 Reed-Muller Codes
Reed-Muller (RM) codes were discovered by Reed and Muller in 1954 [7, 8]. For some integers m and 0 ? r' < m, an r'-th order Reed-Muller code of block length n = 2 m is denoted by R( r', m) and has the following parameters:
-
Block length:
n = 2 m
Information digits:
Number of check digits:
Minimum distance:
d min = 2 m ? r'
Table 3.5 shows the possible ( n, k) Reed-Muller codes that can be generated by r' and m for r' = 0, 1, . . ., 4 and m = 2, 3, . . ., 5. R(1, 5) first-order Reed-Muller code of rate-6/32 was used to send black-and-white photographs from the Mariner 9 spaceprobe to Earth in 1972. The code has a
| m = 2 | 3 | 4 | 5 | |
|---|---|---|---|---|
| r' = 0 | (4, 1) | (8, 1) | (16, 1) | (32, 1) |
| 1 | (4, 3) | (8, 4) | (16, 5) | (32, 6) |
| 2 | (8, 7) | (16, 11) | (32, 16) | |
| 3 | (16, 15) | (32, 26) | ||
| 4 | (32, 31) |
Hamming distance of 2 m ? r' = 16 and can correct 7 errors. A limited number of codewords inhabit the transmission of color pictures.
An r'-th order Reed-Muller code can be constructed...
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