1D Repetition Minimum-Weight Decoder

The 1D repetition decoder is a small classical decoder used to connect parity syndromes with minimum-weight correction chains. It is independent of the 3-qubit state-vector recovery helpers and works directly on binary error and syndrome vectors.

Syndrome Convention

For a length-n binary error chain e, the repetition-code syndrome has length n - 1 and is defined by neighboring parity differences:

s(k) = e(k) xor e(k + 1)

This syndrome records domain walls in the error chain.

Decoder Behavior

decode_repetition_min_weight(s) reconstructs two compatible chains:

• one assuming the first error bit is 0,
• one assuming the first error bit is 1.

It returns whichever compatible chain has lower Hamming weight. Ties are broken in favor of the chain that starts with 0.

Example:

s = [1 1 0];
ehat = decode_repetition_min_weight(s);

The result is a binary correction chain whose neighboring parities reproduce s.

Relationship To QEC

This is the classical decoding problem behind repetition-code correction:

1. parity checks reveal where neighboring bits disagree, 2. the decoder chooses a low-weight error pattern compatible with that syndrome, 3. applying that correction removes the inferred error chain.

For the tiny 3-qubit bit-flip code, correct_bitflip(...) uses a direct syndrome-to-qubit lookup. decode_repetition_min_weight(...) is the more general variable-length version of the same minimum-weight idea.

Main Files

• src/decode_repetition_min_weight.m
• src/binary_syndrome_index.m
• src/decode_majority.m

Examples

The decoder is exercised by the text examples and simulation tests:

octave --no-gui examples/run_text_examples.m
octave --no-gui tests/run_all_tests.m

Tests

The main checks live in:

• tests/test_decoders_and_simulations.m

Run them through:

octave --no-gui tests/run_all_tests.m

Current Limits

decoder.

syndromes.

weight.

• The decoder is a 1D binary repetition decoder, not a general stabilizer-code
• It assumes independent binary chain errors and nearest-neighbor parity
• It does not use probabilities or soft information; it only minimizes Hamming