bchgenpoly

Produce generator polynomials for BCH code

Syntax

genpoly = bchgenpoly(N,K)

genpoly = bchgenpoly(N,K,prim_poly)

genpoly = bchgenpoly(N,K,prim_poly,outputFormat)

[genpoly,T] = bchgenpoly(___)

Description

genpoly = bchgenpoly(N,K) returns the narrow-sense generator polynomial of a BCH code with codeword length N and message length K. For more information, see Generator Polynomial of a BCH Code.

genpoly = bchgenpoly(N,K,prim_poly) also specifies the primitive polynomial.

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

[genpoly,T] = bchgenpoly(___) also returns T, the error-correction capability of the code when using any of the previous syntaxes.

Examples

Create a BCH Generator Polynomial

Create two BCH generator polynomials based on different primitive polynomials.

Set the codeword and message lengths, n and k.

n = 15;
k = 11;

Create the generator polynomial and return the error correction capability, t.

[genpoly,t] = bchgenpoly(15,11)

 
genpoly = GF(2) array. 
 
Array elements = 
 
   1   0   0   1   1

t = 
1

Create a generator polynomial for a (15,11) BCH code using a different primitive polynomial expressed as a character vector. Note that genpoly2 differs from genpoly, which uses the default primitive.

genpoly2 = bchgenpoly(15,11,'D^4 + D^3 + 1')

 
genpoly2 = GF(2) array. 
 
Array elements = 
 
   1   1   0   0   1

Input Arguments

N — Codeword length

integer

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

Example: 15 for M=4

K — Message length

integer

Message length, specified as an integer. N and K must produce a narrow-sense BCH code. To generate the list of valid (N,K) pairs along with the corresponding values of the error-correction capability, run bchnumerr(N). For more information, see Limitations.

Example: 5 specifies a Galois array with five elements

prim_poly — Primitive polynomial

character vector | string scalar | positive integer

Primitive polynomial, specified as:

• A polynomial character vector or string scalar that represents the polynomial by listing the coefficients. For more information, see Representation of Polynomials in Communications Toolbox.

• A positive integer whose binary representation indicates the coefficients of the primitive polynomial. For more information, see Default Primitive Polynomials.

Example: 'D^4+D+1' specifies the primitive polynomial D4+D+1.

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

outputFormat — Output data type

'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 coefficient

row vector

Generator polynomial coefficients, returned as a row vector that represents the coefficients of the narrow-sense generator polynomial of an [N,K] BCH code in order of descending powers. Specify the output datatype with the outputFormat property.

Data Types: gf | double

T — Error correction capability

positive integer

Error correction capability, returned as a positive integer.

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)].

Algorithms

For a description of Bose–Chaudhuri–Hocquenghem (BCH) coding, see [1]. Although bchgenpoly performs intermediate computations in GF(N + 1), the output form is a Galois vector in GF(2).

Generator Polynomial of a BCH Code

The narrow-sense generator polynomial of a BCH code with codeword length N and message length K is LCM[m₁(x), m₂(x), ..., m_2T(x)], where:

• LCM represents the least common multiple,

• m_i(x) represents the minimum polynomial corresponding to αⁱ, α is a root of the default primitive polynomial for the field GF(N + 1),

• T represents the error-correcting capability of the code.

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

References

[1] Peterson, W. Wesley, and E. J. Weldon. Error-Correcting Codes. 2d ed. MIT Press, 1972.

Version History

Introduced before R2006a