Error-Control Block Codes for Communications Engineers

Appendix B presents a list of GF(2 m) tables generated by a binary primitive polynomial p( x) = x m + p m ?1 x m ?1 + p m ?2 x m ?2 + ... + p 1 x + p 0 of degree m, where p j = 0 and 1, 0 ? j ? m ? 1, and 2 ? m ? 6. Each element in GF(2 m) is expressed as a power of some primitive element ? and as a linear combination of ? 0, ? 1, ..., ? m ?1. The polynomial representation of the element ? i, i = 0, 1, ..., 2 m ? 2 is expressed as ? i = a m ?1 ? m ?1 + a m ?2 ? m ?2 + ... + a 0 with binary coefficients and the coefficients of the polynomial representation of ? i is expressed as a binary m-tuple [ a 0 a 1 ... a m ?1].
| Elements | 2-Tuple |
|---|---|
| 0 = 0 | 0 0 |
| 1 = 1 | 1 0 |
| ? |