galois.ReedSolomon.encode

galois.ReedSolomon.encode(message: ArrayLike, output: 'codeword' | 'parity' = 'codeword') → FieldArray

Encodes the message $m$ into the codeword $c$ .

Shortened codes

For the shortened $[n - s, k - s, d]$ code (only applicable for systematic codes), pass $k - s$ symbols into encode() to return the $n - s$ -symbol message.

Parameters:¶

message: ArrayLike¶: The message as either a $k$ -length vector or $(N, k)$ matrix, where $N$ is the number of messages. For systematic codes, message lengths less than $k$ may be provided to produce shortened codewords.
output: 'codeword' | 'parity' = 'codeword'¶: Specify whether to return the codeword or parity symbols only. The default is "codeword".

Returns:¶

If output="codeword", the codeword as either a $n$ -length vector or $(N, n)$ matrix. If output="parity", the parity symbols as either a $n - k$ -length vector or $(N, n - k)$ matrix.

Notes¶

The message vector $m$ is a member of $GF (q)^{k}$ . The corresponding message polynomial $m (x)$ is a degree- $k$ polynomial over $GF (q)$ .

m = [m_{k - 1}, \dots, m_{1}, m_{0}] \in GF (q)^{k}

m (x) = m_{k - 1} x^{k - 1} + \dots + m_{1} x + m_{0} \in GF (q) [x]

The codeword vector $c$ is a member of $GF (q)^{n}$ . The corresponding codeword polynomial $c (x)$ is a degree- $n$ polynomial over $GF (q)$ .

c = [c_{n - 1}, \dots, c_{1}, c_{0}] \in GF (q)^{n}

c (x) = c_{n - 1} x^{n - 1} + \dots + c_{1} x + c_{0} \in GF (q) [x]

The codeword vector is computed by matrix multiplication of the message vector with the generator matrix. The equivalent polynomial operation is multiplication of the message polynomial with the generator polynomial.

c = m G

c (x) = m (x) g (x)

Examples¶

Vector Vector (shortened) Matrix Matrix (shortened)

Encode a single message using the $RS (15, 9)$ code.

In [1]: rs = galois.ReedSolomon(15, 9)

In [2]: GF = rs.field

In [3]: m = GF.Random(rs.k); m
Out[3]: GF([ 8,  9, 13,  5,  1,  5, 14,  5,  2], order=2^4)

In [4]: c = rs.encode(m); c
Out[4]: GF([ 8,  9, 13,  5,  1,  5, 14,  5,  2, 14,  7, 13, 15, 12, 14], order=2^4)

Compute the parity symbols only.

In [5]: p = rs.encode(m, output="parity"); p
Out[5]: GF([14,  7, 13, 15, 12, 14], order=2^4)

Encode a single message using the shortened $RS (11, 5)$ code.

In [6]: rs = galois.ReedSolomon(15, 9)

In [7]: GF = rs.field

In [8]: m = GF.Random(rs.k - 4); m
Out[8]: GF([12,  0,  7, 13,  3], order=2^4)

In [9]: c = rs.encode(m); c
Out[9]: GF([12,  0,  7, 13,  3,  7,  2, 11, 13, 14,  5], order=2^4)

Compute the parity symbols only.

In [10]: p = rs.encode(m, output="parity"); p
Out[10]: GF([ 7,  2, 11, 13, 14,  5], order=2^4)

Encode a matrix of three messages using the $RS (15, 9)$ code.

In [11]: rs = galois.ReedSolomon(15, 9)

In [12]: GF = rs.field

In [13]: m = GF.Random((3, rs.k)); m
Out[13]: 
GF([[ 8, 12, 11,  1, 10,  0,  2, 13, 10],
    [ 8,  5, 10, 10, 13,  8,  0,  9, 12],
    [11,  2,  3,  1,  7,  5, 14,  8,  2]], order=2^4)

In [14]: c = rs.encode(m); c
Out[14]: 
GF([[ 8, 12, 11,  1, 10,  0,  2, 13, 10,  0, 11,  6,  2, 11,  2],
    [ 8,  5, 10, 10, 13,  8,  0,  9, 12,  0, 12, 11,  5,  5,  6],
    [11,  2,  3,  1,  7,  5, 14,  8,  2,  9, 15,  7, 13,  8, 10]],
   order=2^4)

Compute the parity symbols only.

In [15]: p = rs.encode(m, output="parity"); p
Out[15]: 
GF([[ 0, 11,  6,  2, 11,  2],
    [ 0, 12, 11,  5,  5,  6],
    [ 9, 15,  7, 13,  8, 10]], order=2^4)

Encode a matrix of three messages using the shortened $RS (11, 5)$ code.

In [16]: rs = galois.ReedSolomon(15, 9)

In [17]: GF = rs.field

In [18]: m = GF.Random((3, rs.k - 4)); m
Out[18]: 
GF([[ 1,  6,  0,  2,  9],
    [ 0, 10,  9,  7, 10],
    [ 3,  4,  8, 13,  7]], order=2^4)

In [19]: c = rs.encode(m); c
Out[19]: 
GF([[ 1,  6,  0,  2,  9,  9,  9, 14, 14,  3,  4],
    [ 0, 10,  9,  7, 10, 13,  1, 10,  2,  8,  9],
    [ 3,  4,  8, 13,  7,  0,  5,  3,  0,  6,  5]], order=2^4)

Compute the parity symbols only.

In [20]: p = rs.encode(m, output="parity"); p
Out[20]: 
GF([[ 9,  9, 14, 14,  3,  4],
    [13,  1, 10,  2,  8,  9],
    [ 0,  5,  3,  0,  6,  5]], order=2^4)