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# 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 2m-1 for some integer m between 3 and 16.

, 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 - Alpha1)(X - Alpha2)...(X - Alpha2t) where:

• Alpha represents a root of the default primitive polynomial for the field GF(n+1),

• and t represents 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 Alpha 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 - Alphab)(X - Alphab+1)...(X - Alphab+2t-1), where:

• b is an integer,

• Alpha is a root of prim_poly,

• and t is the code's error-correction capability, (n-k)/2.

[genpoly,t] = rsgenpoly(...) returns t, the 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)
g2 = rsgenpoly(7,3,13) % Use nondefault primitive polynomial.
g3 = rsgenpoly(7,3,13,3) % Use b = 3.```

The output is below.

```g = GF(2^3) array. Primitive polynomial = D^3+D+1 (11 decimal)

Array elements =

1     3     1     2     3

g2 = GF(2^3) array. Primitive polynomial = D^3+D^2+1 (13 decimal)

Array elements =

1     4     5     1     5

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

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