encode

R2026b

Block encoder

Syntax

code = encode(msg,n,k)

code = encode(msg,n,k,codingMethod,prim_poly)

code = encode(msg,n,k,codingMethod,genmat)

code = encode(msg,n,k,codingMethod,genpoly)

[code,added] = encode(___)

Description

code = encode(msg,n,k) encodes message, msg, using the Hamming encoding method with codeword length, n, and message length, k.

code = encode(msg,n,k,codingMethod,prim_poly) encodes msg using codingMethod as the encoding method, and prim_poly as the primitive polynomial. This syntax applies when codingMethod is "hamming/binary" or "hamming/decimal".

code = encode(msg,n,k,codingMethod,genmat) encodes msg using codingMethod as the encoding method and genmat as the generator matrix. This syntax applies when codingMethod is "linear/binary" or "linear/decimal".

code = encode(msg,n,k,codingMethod,genpoly) encodes msg using codingMethod as the systematic cyclic code and genpoly, as the generator polynomial. This syntax applies when codingMethod is "cyclic/binary" or "cyclic/decimal".

[code,added] = encode(___) returns the additional variable added. added denotes the number of zeros appended at the end of the message matrix before encoding. You can specify any of the input argument combinations from the previous syntaxes.

Examples

Encode and Decode Message with Hamming Code

Set the values of the codeword length and message length.

n = 15; % Codeword length
k = 11; % Message length

Create a random binary message with the length equal to the message length.

data = randi([0 1],k,1);

Encode the message.

encData = encode(data,n,k,'hamming/binary');

Corrupt the encoded message by introducing an error at a random location.

errLoc = randerr(1,n);
encData = mod(encData + errLoc',2);

Decode the corrupted sequence. Observe that the decoder has correctly recovered the message.

decData = decode(encData,n,k,'hamming/binary');
numerr = biterr(data,decData)

numerr = 
0

Encode and Decode Message with Linear Block Code

Set the values of the codeword length and message length.

n = 7; % Codeword length
k = 3; % Message length

Create a random binary message with the length equal to the message length.

data = randi([0 1],k,1);

Create a cyclic generator polynomial. Then, create a parity-check matrix and convert it into a generator matrix.

pol = cyclpoly(n,k);
parmat = cyclgen(n,pol);
genmat = gen2par(parmat);

Encode the message sequence by using the generator matrix.

encData = encode(data,n,k,'linear/binary',genmat);

Corrupt the encoded message by introducing an error at a random location.

errLoc = randerr(1,n);
encData = mod(encData + errLoc',2);

Decode the corrupted sequence. Observe that the decoder has correctly recovered the message.

decData = decode(encData,n,k,'linear/binary',genmat);

Single-error patterns loaded in decoding table.  8 rows remaining.
2-error patterns loaded.  1 rows remaining.
3-error patterns loaded.  0 rows remaining.

numerr = biterr(data,decData)

numerr = 
0

Encode and Decode Message with Cyclic Block Code

Set the values of the codeword length and message length.

n = 15; % Codeword length
k = 5; % Message length

Create a random binary message with the length equal to the message length.

data = randi([0 1],k,1);

Create a generator polynomial for a cyclic code. Create a parity-check matrix by using the generator polynomial.

genpoly = cyclpoly(n,k);
parmat = cyclgen(n,genpoly);

Create a syndrome decoding table by using the parity-check matrix.

trt = syndtable(parmat);

Single-error patterns loaded in decoding table.  1008 rows remaining.
2-error patterns loaded.  918 rows remaining.
3-error patterns loaded.  648 rows remaining.
4-error patterns loaded.  243 rows remaining.
5-error patterns loaded.  0 rows remaining.

Encode the data by using the generator polynomial.

encData = encode(data,n,k,'cyclic/binary',genpoly);

Corrupt the encoded message by introducing three errors at random locations.

errLoc = randerr(1,n,3);
encData = mod(encData + errLoc',2);

Decode the corrupted sequence. Observe that the decoder has correctly recovered the message.

decData = decode(encData,n,k,'cyclic/binary',genpoly,trt);
numerr = biterr(data,decData)

numerr = 
0

Input Arguments

msg — Input messages

binary column or row vector | binary matrix with k columns | column or row vector of integers in the range [0, 2^k – 1]

Input messages, specified as one of these options:

• Binary column or row vector with k columns

• Binary matrix with k columns

• Column or row vector of k columns and having integers in the range [0, 2^k – 1]

Example: msg = [0 1 1 0, 0 1 0 1, 1 0 0 1] specifies a binary row vector for k = 4.

Example: msg = [0 1 1 0; 0 1 0 1; 1 0 0 1] specifies a binary matrix for k = 4.

Example: msg = [6, 10, 9] specifies a row vector of integers for k = 4.

Data Types: double

n — Codeword length

positive integer

Codeword length, specified as a positive integer such that n = 2^m – 1, where m is an integer greater than or equal to 2.

Data Types: double

k — Message length

positive integer

Message length, specified as a positive integer such that k = n – m, where m is an integer greater than or equal to 2.

Data Types: double

codingMethod — Error coding method and format

"hamming/binary" (default) | "hamming/decimal" | "linear/binary" | ...

Error coding method and format, specified as one of these:

• "hamming/binary"

• "hamming/decimal"

• "linear/binary"

• "linear/decimal"

• "cyclic/binary"

• "cyclic/decimal"

Data Types: char | string

prim_poly — Primitive polynomial

gfprimdf(n-k) (default) | binary row vector | character vector | string scalar | positive integer

Primitive polynomial, specified as one of these options:

• Binary row vector — This vector gives coefficients of prim_poly in the order of ascending powers.

• Character vector or a string scalar — This value defines prim_poly in textual representation. For more information, see Representation of Polynomials in Communications Toolbox.

• Positive integer — This value defines prim_poly in the range [2^m + 1, 2^{m + 1} – 1].

For more information about default primitive polynomials, see Default Primitive Polynomials. For more information about specifying your own primitive polynomial when you create a Galois field array, see Primitive Polynomials and Element Representations.

Dependencies

This input applies when codingMethod is "hamming/binary" or "hamming/decimal".

Data Types: double | char | string

genmat — Generator matrix

k-by-n numeric matrix

Generator matrix, specified as a k-by-n numeric matrix.

Dependencies

This input applies when codingMethod is "linear/binary" or "linear/decimal".

Data Types: double

genpoly — Generator polynomial

cyclpoly(n-k) (default) | binary row vector | character vector | string scalar

Generator polynomial, specified as a polynomial character vector or a row vector that gives the coefficients in order of ascending powers of the binary generator polynomial. The value of genpoly for an [n, k] cyclic code must have degree n–k and divide xⁿ-1, where x is an identifier.

Dependencies

This input applies when codingMethod is "cyclic/binary" or "cyclic/decimal".

Data Types: char | string

Output Arguments

code — Output code

binary column or row vector | binary matrix with n columns | column or row vector of integers in the range [0, 2ⁿ–1].

Output code, returned as one of the options in this table. The value and dimension of code depends on the value and dimension of the msg and the input message format according to this table:

msg Value	Input Message Format	code Value
Binary column or row vector	binary	Binary column or row vector
Binary matrix with k columns	binary	Binary matrix with n columns
Column or row vector of integers in the range [0, 2k–1]	decimal	Column or row vector of integers in the range [0, 2n–1]

added — Additional variable

nonnegative integer

Additional variable, returned as the number of zeros that were appended at the end of the message matrix before encoding for the matrix to have the appropriate size. The size of the message matrix depends on the n, k, and msg and the encoding method.

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

Algorithms

Depending on the error-correction coding method, the encode function relies on lower-level functions such as hammgen and cyclgen.

Version History

Introduced before R2006a