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([15,  9, 15, 13, 10,  8, 13,  2, 14], order=2^4)

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

Compute the parity symbols only.

In [5]: p = rs.encode(m, parity_only=True); p
Out[5]: GF([14, 14, 10,  8,  5,  0], 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([0, 3, 9, 3, 3], order=2^4)

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

Compute the parity symbols only.

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

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

Compute the parity symbols only.

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

In [19]: c = rs.encode(m); c
Out[19]: 
GF([[ 9,  9,  5,  0,  3,  0,  0, 13,  7, 14,  8],
    [ 3,  2,  0,  4,  4,  9,  8,  4,  3,  8,  0],
    [13, 10,  6, 10, 14, 12,  4, 13,  6,  3, 11]], order=2^4)

Compute the parity symbols only.

In [20]: p = rs.encode(m, parity_only=True); p
Out[20]: 
GF([[ 0,  0, 13,  7, 14,  8],
    [ 9,  8,  4,  3,  8,  0],
    [12,  4, 13,  6,  3, 11]], order=2^4)