-
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([ 4, 10, 3, 15, 6, 10, 7, 10, 12], order=2^4) In [4]: c = rs.encode(m); c Out[4]: GF([ 4, 10, 3, 15, 6, 10, 7, 10, 12, 0, 9, 9, 3, 13, 11], order=2^4)
Compute the parity symbols only.
In [5]: p = rs.encode(m, parity_only=True); p Out[5]: GF([ 0, 9, 9, 3, 13, 11], 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([14, 3, 6, 1, 8], order=2^4) In [9]: c = rs.encode(m); c Out[9]: GF([14, 3, 6, 1, 8, 0, 10, 2, 14, 5, 4], order=2^4)
Compute the parity symbols only.
In [10]: p = rs.encode(m, parity_only=True); p Out[10]: GF([ 0, 10, 2, 14, 5, 4], 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([[ 5, 2, 3, 2, 8, 8, 9, 6, 1], [15, 2, 0, 15, 9, 5, 3, 6, 6], [11, 12, 2, 5, 7, 11, 5, 11, 13]], order=2^4) In [14]: c = rs.encode(m); c Out[14]: GF([[ 5, 2, 3, 2, 8, 8, 9, 6, 1, 6, 9, 5, 12, 3, 7], [15, 2, 0, 15, 9, 5, 3, 6, 6, 10, 7, 7, 11, 8, 6], [11, 12, 2, 5, 7, 11, 5, 11, 13, 4, 5, 13, 0, 1, 9]], order=2^4)
Compute the parity symbols only.
In [15]: p = rs.encode(m, parity_only=True); p Out[15]: GF([[ 6, 9, 5, 12, 3, 7], [10, 7, 7, 11, 8, 6], [ 4, 5, 13, 0, 1, 9]], 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([[12, 11, 11, 12, 15], [ 3, 12, 7, 12, 6], [15, 4, 0, 10, 7]], order=2^4) In [19]: c = rs.encode(m); c Out[19]: GF([[12, 11, 11, 12, 15, 0, 9, 3, 9, 7, 11], [ 3, 12, 7, 12, 6, 11, 1, 1, 1, 7, 13], [15, 4, 0, 10, 7, 15, 9, 5, 14, 12, 1]], order=2^4)
Compute the parity symbols only.
In [20]: p = rs.encode(m, parity_only=True); p Out[20]: GF([[ 0, 9, 3, 9, 7, 11], [11, 1, 1, 1, 7, 13], [15, 9, 5, 14, 12, 1]], order=2^4)