cosets

R2026b

Produce cyclotomic cosets for Galois field

Syntax

cst = cosets(m)

cst = cosets(m,prim_poly)

Description

cst = cosets(m) produces cyclotomic cosets mod(2^m – 1).

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 [1] and [2].

cst = cosets(m,prim_poly) specifies the primitive polynomial of the cosets.

Examples

Display Cyclotomic Cosets for GF(8)

Find and display the cyclotomic cosets for $GF(8)$. As an example of interpreting the results, c{2} indicates that $A$, $A^2$, and $A^2+A$ share the same minimal polynomial, where $A$ is a primitive element for $GF(8)$.

c = cosets(3);

 
Primitive polynomial(s) = 
 
D^3+D^1+1

c{1}'

 
ans = GF(2^3) array. Primitive polynomial = D^3+D+1 (11 decimal)
 
Array elements = 
 
   1

c{2}'

 
ans = GF(2^3) array. Primitive polynomial = D^3+D+1 (11 decimal)
 
Array elements = 
 
   2   4   6

c{3}'

 
ans = GF(2^3) array. Primitive polynomial = D^3+D+1 (11 decimal)
 
Array elements = 
 
   3   5   7

Input Arguments

m — Field order

positive integer

Field order, specified as a positive integer from 1 through 16. The function uses this value to calculate the distinct number of elements in the GF.

Data Types: single | double

prim_poly — Primitive polynomial

primitive polynomial in GF(2^m) (default) | binary row vector | character vector | string scalar | positive integer

Primitive polynomial, specified as one of these options:

• Binary row vector — This vector specifies coefficients of prim_poly in the order of ascending powers.

• Character vector or a string scalar — This value defines prim_poly in a textual representation. For more details, refer to Representation of Polynomials in Communications Toolbox.

• Positive integer — This value defines prim_poly in the range [(2m + 1), (2^m+1 – 1)].

If prim_poly is not specified, see Default Primitive Polynomials for the list of default primitive polynomial used for each Galois field array GF(2^m).

Data Types: double | char | string

Output Arguments

cst — Cyclotomic coset

Galois array

Cyclotomic coset, returned as a cell array. Each element of cst is a Galois array that represents one cyclotomic coset.

References

[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.

Version History

Introduced before R2006a