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 = rsgenpoly(N,K,prim_poly,B,outputFormat)

[genpoly,T] = rsgenpoly(___)

Description

genpoly = rsgenpoly(N,K) returns the narrow-sense generator polynomial of an [N,K] Reed-Solomon code. The output genpoly is a Galois field array that represents the coefficients of the generator polynomial in order of descending powers. A narrow-sense BCH code is a BCH code with B = 1. Here, the narrow-sense generator polynomial is (X – α¹)(X – α²)...(X – α^N–K), where α is a root of the default primitive polynomial for the field GF(N+1). For additional information, see Narrow-Sense BCH Codes and Reed-Solomon Codes.

genpoly = rsgenpoly(N,K,prim_poly) also specifies the primitive polynomial, prim_poly, for GF(N+1) that has α as a root.

genpoly = rsgenpoly(N,K,prim_poly,B) returns the generator polynomial, (X – α^B)(X – α^B+1)...(X – α^{B+N–K – 1}), where B is an integer.

genpoly = rsgenpoly(N,K,prim_poly,B,outputFormat) specifies the output format of genpoly as a Galois field array or double-precision array.

[genpoly,T] = rsgenpoly(___) also returns the error-correction capability of the [N,K] Reed-Solomon code, T, using any of the preceding input argument syntaxes.

Examples

Create Narrow-Sense Generator Polynomial

Specify the codeword length, n, and message length, k.

n = 7;
k = 3;

Create the narrow-sense generator polynomial for the [n,k] Reed-Solomon code. genpoly is a Galois field array, by default, that represents the coefficients of this generator polynomial in order of descending powers.

genpoly = rsgenpoly(n,k)

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

Create Narrow-Sense Generator Polynomial Specifying Primitive Polynomial

Create the narrow-sense generator polynomial of a Reed-Solomon code with respect to the primitive polynomial ${\mathit{D}}^3+{\mathit{D}}^2+1$.

Specify the codeword length, n, message length, k, and primitive polynomial ${\mathit{D}}^3+{\mathit{D}}^2+1$ represented in decimal form.

n = 7;
k = 3;
prim_poly = 13;

Create the narrow-sense generator polynomial for the [n,k] Reed-Solomon code with respect to primitive polynomial ${\mathit{D}}^3+{\mathit{D}}^2+1$ for GF(8). genpoly is a Galois field array, by default, that represents the coefficients of this generator polynomial in order of descending powers.

genpoly = rsgenpoly(n,k,prim_poly)

 
genpoly = GF(2^3) array. Primitive polynomial = D^3+D^2+1 (13 decimal)
 
Array elements = 
 
   1   4   5   1   5

Create Generator Polynomial for Specified B

Create the generator polynomial of a Reed-Solomon code with respect to the default primitive polynomial.

Specify the codeword length, n, message length, k, and exponent of $\alpha$, b.

n = 7;
k = 3;
b = 4;

Create the generator polynomial $\left(\mathit{X}-{\alpha }^4\right)\left(\mathit{X}-{\alpha }^5\right)\left(\mathit{X}-{\alpha }^6\right)\left(\mathit{X}-{\alpha }^7\right)$, with respect to the default primitive polynomial. genpoly is a Galois field array that represents the coefficients of this generator polynomial in order of descending powers. Display the error-correcting capability of the code.

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

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

t = 
2

Create Generator Polynomial for DVB-S and WiMAX

Create the generator polynomial of a Reed-Solomon code with respect to the primitive polynomial ${\mathit{D}}^8+{\mathit{D}}^4+{\mathit{D}}^3+{\mathit{D}}^2+1$.

Specify the codeword length, n, message length, k, the primitive polynomial represented in decimal form, and the exponent of $\alpha$, b.

n = 255;
k = 239;
prim_poly = 285;
b = 0;

Create the generator polynomial for the [n,k] Reed-Solomon code. genpoly is a Galois field array that represents a generator polynomial and is compliant with DVB-S and WiMAX.

genpoly = rsgenpoly(n,k,prim_poly,b)

 
genpoly = GF(2^8) array. Primitive polynomial = D^8+D^4+D^3+D^2+1 (285 decimal)
 
Array elements = 
 
     1    59    13   104   189    68   209    30     8   163    65    41   229    98    50    36    59

Create Narrow-Sense Generator Polynomial with Output Format Double

Create the narrow-sense generator polynomial of a Reed-Solomon code. Specify the output data type as a double-precision array.

Specify the codeword length, n, and message length, k.

n = 7;
k = 3;

Create the narrow-sense generator polynomial for the [n,k] Reed-Solomon code. genpoly is a double-precision array, that represents the coefficients of this generator polynomial in order of descending powers. Specify defaults values for the primitive polynomial and exponent of $\alpha$ inputs by assigning [ ] for them.

genpoly = rsgenpoly(n,k,[],[],'double')

genpoly = 1×5

     1     3     1     2     3

Determine Corresponding Galois Field Array Value

Use the genpoly2b function to determine the corresponding Galois field array value.

Use rsgenpoly with codeword length of 7 and message word length of 3 to create a valid Galois field array.

n = 7;
k = 3;

genpoly = rsgenpoly(n,k)

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

Use genpoly2b to determine the corresponding Galois field array value for the polynomial input.

b = genpoly2b(genpoly)

b = 
1

Input Arguments

N — Codeword length

positive odd integer

Codeword length, specified as an integer of the form N = 2^M – 1, where M is in the range [3,16]. For more information, see Limitations.

Example: Set N to 15 for M=4.

K — Message length

positive integer

Message length, specified as a positive integer. For more information, see Limitations.

prim_poly — Primitive polynomial

GF(N+1) (default) | positive integer

Primitive polynomial, specified as a positive integer. 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 []. For more information, see Default Primitive Polynomials.

Example: 19 specifies the primitive polynomial D4+D+1 because its binary representation is 10011.

B — Exponent of α

1 (default) | positive integer

Exponent of α, specified as a positive integer. α is a root of prim_poly.

outputFormat — Output format

'gf' (default) | 'double'

Output format of genpoly, specified as:

• 'gf' — to output a Galois field array.

• 'double' — to output a double-precision array of the Galois field values.

For more information, see Working with Galois Fields.

Output Arguments

genpoly — Generator polynomial coefficients

Galois field array | double-precision array

Generator polynomial coefficients in descending order, returned as a Galois field array or double-precision array. genpoly is a row vector that represents the coefficients of the narrow-sense generator polynomial of an [N,K] Reed-Solomon code in order of descending powers.

T — Error-correction capability

positive integer

Error-correction capability of the code, returned as a positive integer equal to ⌊(N – K)/2⌋.

Limitations

• Valid values for N = 2^M – 1, where M is an integer in the range [3,16]. The maximum allowable value of N = 2¹⁶ – 1 = 65,535.

• Valid values for K = [1,N – 1].

More About

Default Primitive Polynomials

This table lists the default primitive polynomial used for each Galois field array GF(2^m). To use a different primitive polynomial, specify prim_poly as an input argument. prim_poly must be in the range [(2^m + 1), (2^m+1 – 1)] and must indicate an irreducible polynomial. For more information, see Primitive Polynomials and Element Representations.

Value of m	Default Primitive Polynomial	Integer Representation
1	D + 1	3
2	D2 + D + 1	7
3	D3 + D + 1	11
4	D4 + D + 1	19
5	D5 + D2 + 1	37
6	D6 + D + 1	67
7	D7 + D3 + 1	137
8	D8 + D4 + D3 + D2 + 1	285
9	D9 + D4 + 1	529
10	D10 + D3 + 1	1033
11	D11 + D2 + 1	2053
12	D12 + D6 + D4 + D + 1	4179
13	D13 + D4 + D3 + D + 1	8219
14	D14 + D10 + D6 + D + 1	17475
15	D15 + D + 1	32771
16	D16 + D12 + D3 + D + 1	69643

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

All inputs must be constants. Expressions or variables are allowed if their values do not change.

Version History

Introduced before R2006a