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# Integer-Input RS Encoder

Create Reed-Solomon code from integer vector data

## Library

Block sublibrary of Error Detection and Correction

## Description

The Integer-Input RS Encoder block creates a Reed-Solomon code with message length, K, and codeword length, (N - number of punctures). You specify both N and K directly in the block dialog. The symbols for the code are integers between 0 and 2M-1, which represent elements of the finite field GF(2M). Restrictions on M and N are described in Restrictions on M and the Codeword Length N below.

This block can output shortened codewords when N and K are appropriately specified. To specify output codewords that are shortened by a length S, N and K must be specified in the dialog box as Nfull – S and Kfull – S, where Nfull and Kfull are the N and K of an unshortened code. If $\text{S}<\left({\text{N}}_{\text{full}}+1\right)/2$, the encoder can automatically determine the value of Nfull and Kfull. However, if $\text{S}\ge \left({\text{N}}_{\text{full}}+1\right)/2$, Primitive polynomial must be specified in order to properly define the extension field for the code.

The input and output are integer-valued signals that represent messages and codewords, respectively. This block accepts a column vector input signal with a length that is an integer multiple of K. The column vector output, with a length that is the same integer multiple of N, inherits its data type from the input signal. For information about the data types each block port supports, see the Supported Data Type table on this page.

For more information on representing data for Reed-Solomon codes, see the section Integer Format (Reed-Solomon Only) in Communications System Toolbox™ User's Guide.

If the encoder is processing multiple codewords per frame, then the same puncture pattern holds for all codewords.

The default value of M is the smallest integer that is greater than or equal to log2(N+1), that is, ceil(log2(N+1)). You can change the value of M from the default by specifying the primitive polynomial for GF(2M), as described in Specifying the Primitive Polynomial below. If N is less than 2M-1, the block uses a shortened Reed-Solomon code.

An (N, K) Reed-Solomon code can correct up to floor((N-K)/2) symbol errors (not bit errors) in each codeword.

### Specifying the Primitive Polynomial

You can specify the primitive polynomial that defines the finite field GF(2M), corresponding to the integers that form messages and codewords. To do so, first select Specify primitive polynomial. Then, in the Primitive polynomial field, enter a binary row vector that represents a primitive polynomial over GF(2) of degree M, in descending order of powers. For example, to specify the polynomial x3+x+1, enter the vector [1 0 1 1].

If you do not select Specify primitive polynomial, the block uses the default primitive polynomial of degree M = ceil(log2(N+1)). You can display the default polynomial by entering primpoly(ceil(log2(N+1))) at the MATLAB® prompt.

### Restrictions on M and the Codeword Length N

The restrictions on the degree M of the primitive polynomial and the codeword length N are as follows:

• If you do not select Specify primitive polynomial, N must lie in the range 7 < N ≤ 216–1.

• If you do select Specify primitive polynomial, N must lie in the range 7 ≤ N ≤ 2M–1 and M must lie in the range 3 ≤ M ≤ 16.

### Specifying the Generator Polynomial

You can specify the generator polynomial for the Reed-Solomon code. To do so, first select Specify generator polynomial. Then, in the Generator polynomial field, enter an integer row vector whose entries are between 0 and 2M-1. The vector represents a polynomial, in descending order of powers, whose coefficients are elements of GF(2M) represented in integer format. See the section Integer Format (Reed-Solomon Only) for more information about integer format. The generator polynomial must be equal to a polynomial with a factored form

g(x) = (x+Ab)(x+Ab+1)(x+Ab+2)...(x+Ab+N-K-1)

where A is the primitive element of the Galois field over which the input message is defined, and b is an integer.

If you do not select Specify generator polynomial, the block uses the default generator polynomial, corresponding to b=1, for Reed-Solomon encoding. You can display the default generator polynomial by entering rsgenpoly(N1,K1), where N1 = 2^M-1 and K1 = K+(N1-N), at the MATLAB prompt, if you are using the default primitive polynomial. If the Specify primitive polynomial box is selected, and you specify the primitive polynomial specified as poly, the default generator polynomial is rsgenpoly(N1,K1,poly).

 Note:   The degree of the generator polynomial, M, must be equal to the codeword length, N, minus the message length, K, that is, M = N − K.

### Puncture Codes

The block supports puncturing when you select the Puncture code parameter. This enables the Puncture vector parameter, which takes in a binary vector to specify the puncturing pattern. For a puncture vector, 1 represents that the data symbol passes unaltered, and 0 represents that the data symbol gets punctured, or removed, from the data stream. This convention is carried for both the encoder and the decoder. For more information, see Shortening, Puncturing, and Erasures.

## Examples

Suppose M = 3, N = 23-1 = 7, and K = 5. Then a message is a vector of length 5 whose entries are integers between 0 and 7. A corresponding codeword is a vector of length 7 whose entries are integers between 0 and 7. The following figure illustrates possible input and output signals to this block when Codeword length N is set to 7, Message length K is set to 5, and the default primitive and generator polynomials are used.

## Dialog Box

Codeword length N

The codeword length.

Message length K

The message length.

Specify primitive polynomial

Selecting this check box enables the Primitive polynomial parameter.

Primitive polynomial

Binary row vector representing the primitive polynomial in descending order of powers. When you provide a Primitive polynomial, the number of input bits must be an integer multiple of K times the order of the Primitive polynomial instead.

This parameter applies when only when you select Specify primitive polynomial.

Specify generator polynomial

Selecting this check box enables the Generator polynomial parameter.

Generator polynomial

This field is available only when Specify generator polynomial is selected.

Integer row vector, whose entries are in the range from 0 to 2M-1, representing the generator polynomial in descending order of powers. Each coefficient is an element of the Galois field defined by the primitive polynomial.

Puncture code

Selecting this check box enables the Puncture vector parameter.

Puncture vector

This field is available only when Puncture code is selected.

A column vector of length N-K. In a puncture vector, 1 represents that the data symbol passes unaltered, and 0 represents that the data symbol gets punctured, or removed, from the data stream.

The default value is [ones(2,1); zeros(2,1)].

This parameter applies only when you select Puncture code.

## Supported Data Type

PortSupported Data Types
In
• Double-precision floating point

• Single-precision floating point

• 8-, 16-, and 32-bit signed integers

• 8-, 16-, and 32-bit unsigned integers

Out
• Double-precision floating point

• Single-precision floating point

• 8-, 16-, and 32-bit signed integers

• 8-, 16-, and 32-bit unsigned integers

## Pair Block

Integer-Output RS Decoder