Communications Toolbox

rsgenpoly

Generator polynomial of Reed-Solomon code

Syntax

• genpoly = rsgenpoly(n,k)
  genpoly = rsgenpoly(n,k,prim_poly)
  genpoly = rsgenpoly(n,k,prim_poly,b)
  [genpoly,t] = rsgenpoly(...)
  

Description

genpoly = rsgenpoly(n,k) returns the narrow-sense generator polynomial of a Reed-Solomon code with codeword length n and message length k. The codeword length n must have the form 2^m-1 for some integer m, and n-k must be an even integer. The output genpoly is a Galois row vector that represents the coefficients of the generator polynomial in order of descending powers. The narrow-sense generator polynomial is (X - A¹)(X - A²)...(X - A^2t) where A is a root of the default primitive polynomial for the field GF(n+1) and t is the code's error-correction capability, (n-k)/2.

genpoly = rsgenpoly(n,k,prim_poly) is the same as the syntax above, except that prim_poly specifies the primitive polynomial for GF(n+1) that has A as a root. prim_poly is an integer whose binary representation indicates the coefficients of the primitive polynomial. To use the default primitive polynomial GF(n+1), set prim_poly to [].

genpoly = rsgenpoly(n,k,prim_poly,b) returns the generator polynomial (X - A^b)(X - A^b+1)...(X - A^b+2t-1) where b is an integer, A is a root of prim_poly and t is the code's error-correction capability, (n-k)/2.

[genpoly,t] = rsgenpoly(...) returns t, the code error-correction capability of the code.

Examples

The examples below create Galois row vectors that represent generator polynomials for a [7,3] Reed-Solomon code. The vectors g and g2 both represent the narrow-sense generator polynomial, but with respect to different primitive elements A. More specifically, g2 is defined such that A is a root of the primitive polynomial D³ + D² + 1 for GF(8), not of the default primitive polynomial D³ + D + 1. The vector g3 represents the generator polynomial (X - A³)(X - A⁴)(X - A⁵)(X - A⁶), where A is a root of D³ + D² + 1 in GF(8).

• g = rsgenpoly(7,3)
   
  g = GF(2^3) array. Primitive polynomial = D^3+D+1 (11 decimal)
   
  Array elements = 
   
       1     3     1     2     3
  
  g2 = rsgenpoly(7,3,13) % Use nondefault primitive polynomial.
   
  g2 = GF(2^3) array. Primitive polynomial = D^3+D^2+1 (13 decimal)
   
  Array elements = 
   
       1     4     5     1     5
  
  g3 = rsgenpoly(7,3,13,3) % Use b = 3.
   
  g3 = GF(2^3) array. Primitive polynomial = D^3+D^2+1 (13 decimal)
   
  Array elements = 
   
       1     7     1     6     7
  

As another example, the command below shows that the default narrow-sense generator polynomial for a [15,11] Reed-Solomon code is X⁴ + (A³ + A² + 1)X³ + (A³ + A²)X² + A³X + (A² + A + 1) where A is a root of the default primitive polynomial for GF(16).

• gp = rsgenpoly(15,11)
  
  gp = GF(2^4) array. Primitive polynomial = D^4+D+1 (19 decimal)
   
  Array elements = 
   
       1    13    12     8     7
  

For additional examples, see Parameters for Reed-Solomon Codes.

Limitations

n and k must differ by an even integer. The maximum allowable value of n is 65535.

See Also

gf, rsenc, rsdec

rsencof

scatterplot