Produce cyclotomic cosets for Galois field
cst = cosets(m)
cst = cosets(m) produces cyclotomic cosets
mod 2^m-1. Each element of the cell array cst
is a Galois array that represents one cyclotomic coset.
A cyclotomic coset is a set of elements that share the same minimal polynomial.
Together, the cyclotomic cosets mod 2^m-1 form a partition of the
group of nonzero elements of GF(2^m). For more details on
cyclotomic cosets, see the works listed in References.
The commands below find and display the cyclotomic cosets for GF(8). As an example
of interpreting the results, c{2} indicates that A,
A2, and A2 + A share the
same minimal polynomial, where A is a primitive element for GF(8).
c = cosets(3);
c{1}'
c{2}'
c{3}'The output is below.
ans = GF(2^3) array. Primitive polynomial = D^3+D+1 (11 decimal)
Array elements =
1
ans = GF(2^3) array. Primitive polynomial = D^3+D+1 (11 decimal)
Array elements =
2 4 6
ans = GF(2^3) array. Primitive polynomial = D^3+D+1 (11 decimal)
Array elements =
3 5 7
[1] Blahut, Richard E., Theory and Practice of Error Control Codes, Reading, MA, Addison-Wesley, 1983, p. 105.
[2] Lin, Shu, and Daniel J. Costello, Jr., Error Control Coding: Fundamentals and Applications, Englewood Cliffs, NJ, Prentice-Hall, 1983.