Communications Toolbox | ![]() ![]() |
Produce cyclotomic cosets for a Galois field
Syntax
cs = gfcosets(m); cs = gfcosets(m,p);
Description
cs = gfcosets(m)
produces the cyclotomic cosets for GF(2m
), where m
is a positive integer.
cs = gfcosets(m,p)
produces the cyclotomic cosets for GF(p
m
), where m
is a positive integer and p
is a prime number.
In both cases, the output matrix cs
is structured so that each row represents one coset. The row represents the coset by giving the exponential format of the elements of the coset, relative to the default primitive polynomial for the field. For a description of exponential formats, see Representing Elements of Galois Fields.
The first column contains the coset leaders. Because the lengths of cosets may vary, entries of NaN
are used to fill the extra spaces when necessary to make cs
rectangular.
A cyclotomic coset is a set of elements that all satisfy the same minimal polynomial. For more details on cyclotomic cosets, see the works listed in References below.
Examples
The command below finds the cyclotomic cosets for GF(9).
cs = gfcosets(2,3) cs = 0 NaN 1 3 2 6 4 NaN 5 7
The gfminpol
function can check that the elements of, for example, the third row of cs
indeed belong in the same coset.
m = [gfminpol(2,2,3); gfminpol(6,2,3)] % Rows are identical. m = 2 0 1 2 0 1
See Also
gfminpol
, gfroots
, gfprimdf
References
Blahut, Richard E. Theory and Practice of Error Control Codes. Reading, Mass.: Addison-Wesley, 1983, p.105.
Lin, Shu and Daniel J. Costello, Jr. Error Control Coding: Fundamentals and Applications. Englewood Cliffs, N.J.: Prentice-Hall, 1983.
![]() | gfconv | gfdeconv | ![]() |