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

Encodes the message m into the Reed-Solomon codeword c.

Parameters
message: ndarray | FieldArray

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. The return type matches the message type. 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 m is defined as m=[mk−1,…,m1,m0]∈GF(q)k, which corresponds to the message polynomial m(x)=mk−1xk−1+⋯+m1x+m0. The codeword vector c is defined as c=[cn−1,…,c1,c0]∈GF(q)n, which corresponds to the codeword polynomial c(x)=cn−1xn−1+⋯+c1x+c0.

The codeword vector is computed from the message vector by c=mG, where G is the generator matrix. The equivalent polynomial operation is c(x)=m(x)g(x). For systematic codes, G=[I | P] such that c=[m | p]. And in polynomial form, p(x)=−(m(x)xn−k mod g(x)) with c(x)=m(x)xn−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 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 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([13, 10, 13,  6, 10,  6, 15,  1,  0], order=2^4)

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

Compute the parity symbols only.

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

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

Compute the parity symbols only.

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

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

Compute the parity symbols only.

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

In [19]: c = rs.encode(m); c
Out[19]: 
GF([[ 8,  3,  5, 11, 13,  8,  3, 13,  6,  8,  2],
    [ 4,  9,  3, 12, 12, 11, 12,  4, 12, 15,  1],
    [ 9,  0,  3,  1, 13,  4, 13,  9,  1, 11,  7]], order=2^4)

Compute the parity symbols only.

In [20]: p = rs.encode(m, parity_only=True); p
Out[20]: 
GF([[ 8,  3, 13,  6,  8,  2],
    [11, 12,  4, 12, 15,  1],
    [ 4, 13,  9,  1, 11,  7]], order=2^4)