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
Appendix C: Minimal Polynomials of Elements in GF(2m )
Appendix C presents a list of minimal polynomials of elements in GF(2 m). The degree of the minimal polynomial ?( x) = ? mx m + ? m ?1 x m ?1 + ... + ? 1 x + ? 0 is m, where ? j = 0 and 1, 0 ? j ? m and 2 ? m ? 6. The finite fields GF(2 2), GF(2 3), GF(2 4), GF(2 5), and GF(2 6) are constructed using the primitive polynomials x 2 + x + 1, x 3 + x + 1, x 4 + x + 1, x 5 + x 2 + 1, and x 6 + x + 1 with ? as a root, respectively.
| m | Conjugate Roots of ?( x) | Order of Element | Minimal Polynomial ?( x) |
|---|---|---|---|
| 2 | ?, ? 2 | 3 | x 2 + x + 1 |
| 3 | ?, ? 2, ? 4 | 7 | x 3 + x + 1 |
| 3 | ? 3, ? 6, ? 5 | 7 | x 3 + x 2 + 1 |
| 4 | ?, ? 2, ? 4, ? 8 |
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