galois.ReedSolomon.encode(message: ArrayLike, parity_only: bool = False) → FieldArray

Encodes the message \(\mathbf{m}\) into the Reed-Solomon codeword \(\mathbf{c}\).

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.

parity_only: bool = False

Optionally specify whether to return only the parity symbols. This only applies to systematic codes. The default is False.

Returns

The codeword as either a \(n\)-length vector or \((N, n)\) matrix. If parity_only=True, the parity symbols are returned as either a \(n - k\)-length vector or \((N, n-k)\) matrix.

Notes

The message vector \(\mathbf{m}\) is defined as \(\mathbf{m} = [m_{k-1}, \dots, m_1, m_0] \in \mathrm{GF}(q)^k\), which corresponds to the message polynomial \(m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0\). The codeword vector \(\mathbf{c}\) is defined as \(\mathbf{c} = [c_{n-1}, \dots, c_1, c_0] \in \mathrm{GF}(q)^n\), which corresponds to the codeword polynomial \(c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0\).

The codeword vector is computed from the message vector by \(\mathbf{c} = \mathbf{m}\mathbf{G}\), where \(\mathbf{G}\) is the generator matrix. The equivalent polynomial operation is \(c(x) = m(x)g(x)\). For systematic codes, \(\mathbf{G} = [\mathbf{I}\ |\ \mathbf{P}]\) such that \(\mathbf{c} = [\mathbf{m}\ |\ \mathbf{p}]\). And in polynomial form, \(p(x) = -(m(x) x^{n-k}\ \textrm{mod}\ g(x))\) with \(c(x) = m(x)x^{n-k} + p(x)\). For systematic and non-systematic codes, each codeword is a multiple of the generator polynomial, i.e. \(g(x)\ |\ c(x)\).

For the shortened \(\textrm{RS}(n-s, k-s)\) code (only applicable for systematic codes), pass \(k-s\) symbols into encode() to return the \(n-s\)-symbol codeword.

Examples

Encode a single message using the \(\textrm{RS}(15, 9)\) code.

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

In [2]: GF = rs.field

In [3]: m = GF.Random(rs.k); m
Out[3]: GF([ 2, 12,  3, 14,  6,  4, 15,  5, 15], order=2^4)

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

Compute the parity symbols only.

In [5]: p = rs.encode(m, parity_only=True); p
Out[5]: GF([13, 14, 13, 10,  9, 15], order=2^4)

Encode a single message using the shortened \(\textrm{RS}(11, 5)\) 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([ 7, 11, 11, 11,  7], order=2^4)

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

Compute the parity symbols only.

In [10]: p = rs.encode(m, parity_only=True); p
Out[10]: GF([8, 5, 2, 1, 0, 3], order=2^4)

Encode a matrix of three messages using the \(\textrm{RS}(15, 9)\) 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([[ 4, 10,  6,  1,  0,  9, 11,  8,  0],
    [ 2,  5,  6, 15, 14,  6,  4,  0, 11],
    [ 6,  6, 10, 12, 10,  0,  4, 14, 10]], order=2^4)

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

Compute the parity symbols only.

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

Encode a matrix of three messages using the shortened \(\textrm{RS}(11, 5)\) 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([[ 5, 12, 11,  0, 15],
    [ 9,  0,  3,  7,  4],
    [ 1,  9,  7,  2,  0]], order=2^4)

In [19]: c = rs.encode(m); c
Out[19]: 
GF([[ 5, 12, 11,  0, 15,  8, 14,  8,  8, 10,  9],
    [ 9,  0,  3,  7,  4, 10,  3, 12,  2, 10, 13],
    [ 1,  9,  7,  2,  0, 11,  3, 10,  8,  0, 12]], order=2^4)

Compute the parity symbols only.

In [20]: p = rs.encode(m, parity_only=True); p
Out[20]: 
GF([[ 8, 14,  8,  8, 10,  9],
    [10,  3, 12,  2, 10, 13],
    [11,  3, 10,  8,  0, 12]], order=2^4)
Last update: Sep 02, 2022