Bug in Reed-Solomon error-erasure decoder

[luan-xiaokun]

Steps To Reproduce

A Reed-Solomon code RSC(n,k) is supposed to correct at most n - k erasures, while the following snippet shows that this error-erasure decoder does not work as expected.

F = GF(59)
n, k = 40, 12
C = codes.GeneralizedReedSolomonCode(F.list()[:n], k)
D = codes.decoders.GRSErrorErasureDecoder(C)
erasure_num = n - k  # work as expected if changed to values less than n - k

message = vector(F, list(range(1, k + 1)))
codeword = C.encode(message)
for i in range(erasure_num):
    codeword[i] = F.from_integer(1)
erasure = vector(GF(2), [1] * erasure_num + [0] * (n - erasure_num))
print(D.decode_to_message((codeword, erasure)))
# prints (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) instead of (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)

Expected Behavior

The decoded message should be (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)

Actual Behavior

It prints (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) instead

Additional Information

I'm not very familiar with coding theory, but it seems that when there are n-k erasures and no errors, the decoding process is essentially solving a linear equation, and the following snippet works for my case.

received = vector(F, [w for i, w in enumerate(received) if erasure[i]])
vandermonde = matrix([[multipliers[i] * eval_points[i] ** j for j in range(cut)] for i in range(cut)])
decoded = vandermonde.inverse() * received

Environment

• OS: Ubuntu 22.04
• Sage Version: 10.4

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide