-
galois.ReedSolomon.encode(message: ArrayLike, output: Literal[codeword] | Literal[parity] =
'codeword'
) FieldArray Encodes the message \(\mathbf{m}\) into the 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.
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.- output: Literal[codeword] | Literal[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. Ifoutput="parity"
, the parity symbols as either a \(n-k\)-length vector or \((N, n-k)\) matrix.
Notes¶
The message vector \(\mathbf{m}\) is a member of \(\mathrm{GF}(q)^k\). The corresponding message polynomial \(m(x)\) is a degree-\(k\) polynomial over \(\mathrm{GF}(q)\).
\[\mathbf{m} = [m_{k-1},\ \dots,\ m_1,\ m_0] \in \mathrm{GF}(q)^k\]\[m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0 \in \mathrm{GF}(q)[x]\]The codeword vector \(\mathbf{c}\) is a member of \(\mathrm{GF}(q)^n\). The corresponding codeword polynomial \(c(x)\) is a degree-\(n\) polynomial over \(\mathrm{GF}(q)\).
\[\mathbf{c} = [c_{n-1},\ \dots,\ c_1,\ c_0] \in \mathrm{GF}(q)^n\]\[c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0 \in \mathrm{GF}(q)[x]\]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.
\[\mathbf{c} = \mathbf{m} \mathbf{G}\]\[c(x) = m(x) g(x)\]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([14, 5, 5, 4, 9, 1, 0, 2, 9], order=2^4) In [4]: c = rs.encode(m); c Out[4]: GF([14, 5, 5, 4, 9, 1, 0, 2, 9, 7, 15, 9, 8, 2, 6], order=2^4)
Compute the parity symbols only.
In [5]: p = rs.encode(m, output="parity"); p Out[5]: GF([ 7, 15, 9, 8, 2, 6], 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, 13, 4, 10, 5], order=2^4) In [9]: c = rs.encode(m); c Out[9]: GF([ 7, 13, 4, 10, 5, 9, 1, 11, 4, 9, 4], order=2^4)
Compute the parity symbols only.
In [10]: p = rs.encode(m, output="parity"); p Out[10]: GF([ 9, 1, 11, 4, 9, 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([[11, 9, 13, 0, 14, 6, 1, 6, 15], [ 5, 9, 7, 10, 11, 1, 5, 9, 12], [10, 1, 11, 3, 2, 15, 7, 5, 10]], order=2^4) In [14]: c = rs.encode(m); c Out[14]: GF([[11, 9, 13, 0, 14, 6, 1, 6, 15, 11, 14, 5, 13, 12, 9], [ 5, 9, 7, 10, 11, 1, 5, 9, 12, 14, 6, 14, 9, 3, 3], [10, 1, 11, 3, 2, 15, 7, 5, 10, 0, 13, 14, 10, 4, 11]], order=2^4)
Compute the parity symbols only.
In [15]: p = rs.encode(m, output="parity"); p Out[15]: GF([[11, 14, 5, 13, 12, 9], [14, 6, 14, 9, 3, 3], [ 0, 13, 14, 10, 4, 11]], 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([[ 3, 6, 15, 5, 9], [ 6, 10, 9, 12, 3], [ 3, 15, 3, 0, 4]], order=2^4) In [19]: c = rs.encode(m); c Out[19]: GF([[ 3, 6, 15, 5, 9, 9, 7, 13, 4, 0, 10], [ 6, 10, 9, 12, 3, 14, 9, 10, 4, 10, 9], [ 3, 15, 3, 0, 4, 10, 2, 3, 0, 7, 5]], order=2^4)
Compute the parity symbols only.
In [20]: p = rs.encode(m, output="parity"); p Out[20]: GF([[ 9, 7, 13, 4, 0, 10], [14, 9, 10, 4, 10, 9], [10, 2, 3, 0, 7, 5]], order=2^4)