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FA-72176 / Error-correcting codes / Open access

QR format decoder only accepts two bit errors · case 01

Format words with three bit errors are rejected although the code corrects them.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The acceptance radius is dist < 3.

THE FAILURE

The acceptance radius is dist < 3.

Unsuccessful approach: Accepting up to distance 4 can decode a word equidistant from two format words.

Case contract

QR code format information, a BCH(15,5) code. "encode" takes [level, mask] with level bits L=01, M=00, Q=11, H=10 and mask 0..7 (else None): data = level<<3 | mask, append the 10-bit remainder of data*x^10 modulo 0x537, then XOR 0x5412. "decode" takes a 15-bit integer and returns [level, mask, distance] for the unique format word within Hamming distance 3, else None.

Why this case matters

QR readers must recover the error-correction level and mask pattern before decoding any data module.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(op, x):
    EC = {'L': 1, 'M': 0, 'Q': 3, 'H': 2}
    def enc(level, mask):
        data = (EC[level] << 3) | mask
        rem = data << 10
        for i in range(14, 9, -1):
            if rem & (1 << i):
                rem ^= 0x537 << (i - 10)
        return ((data << 10) | rem) ^ 0x5412
    if op == 'encode':
        level, mask = x
        if level not in EC or not 0 <= mask <= 7:
            return None
        return enc(level, mask)
    best = None
    for level in 'LMQH':
        for mask in range(8):
            dist = bin(enc(level, mask) ^ x).count('1')
            if dist < 3 and (best is None or dist < best[2]):
                best = [level, mask, dist]
    return best
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression ["decode", 32647]', ['decode', 32647], ['M', 6, 3]], ['regression ["decode", 1588]', ['decode', 1588], ['Q', 4, 3]], ['partial-repair ["decode", 21533]', ['decode', 21533], None], ['control ["encode", ["L", 0]]', ['encode', ['L', 0]], 30660], ['control ["encode", ["L", 3]]', ['encode', ['L', 3]], 30877], ['control ["encode", ["L", 5]]', ['encode', ['L', 5]], 25368], ['control ["encode", ["L", 7]]', ['encode', ['L', 7]], 26998], ['control ["encode", ["M", 0]]', ['encode', ['M', 0]], 21522]], [['regression ["decode", 31121]', ['decode', 31121], ['L', 3, 3]], ['regression ["decode", 2309]', ['decode', 2309], ['H', 6, 3]], ['partial-repair ["decode", 21533]', ['decode', 21533], None], ['control ["encode", ["M", 3]]', ['encode', ['M', 3]], 23371], ['control ["encode", ["M", 5]]', ['encode', ['M', 5]], 16590], ['control ["encode", ["M", 7]]', ['encode', ['M', 7]], 19104], ['control ["encode", ["Q", 0]]', ['encode', ['Q', 0]], 13663], ['control ["encode", ["Q", 3]]', ['encode', ['Q', 3]], 14854]], [['regression ["decode", 32647]', ['decode', 32647], ['M', 6, 3]], ['regression ["decode", 1588]', ['decode', 1588], ['Q', 4, 3]], ['partial-repair ["decode", 21533]', ['decode', 21533], None], ['control ["encode", ["Q", 5]]', ['encode', ['Q', 5]], 8579], ['control ["encode", ["Q", 7]]', ['encode', ['Q', 7]], 11245], ['control ["encode", ["H", 0]]', ['encode', ['H', 0]], 5769], ['control ["encode", ["H", 3]]', ['encode', ['H', 3]], 6608], ['control ["encode", ["H", 5]]', ['encode', ['H', 5]], 597]], [['regression ["decode", 31121]', ['decode', 31121], ['L', 3, 3]], ['regression ["decode", 2309]', ['decode', 2309], ['H', 6, 3]], ['partial-repair ["decode", 21533]', ['decode', 21533], None], ['control ["encode", ["H", 7]]', ['encode', ['H', 7]], 2107], ['control ["decode", 19724]', ['decode', 19724], ['H', 6, 1]], ['control ["decode", 6616]', ['decode', 6616], ['H', 3, 1]], ['control ["decode", 30660]', ['decode', 30660], ['L', 0, 0]], ['control ["decode", 2099]', ['decode', 2099], ['H', 7, 1]]], [['regression ["decode", 32647]', ['decode', 32647], ['M', 6, 3]], ['regression ["decode", 1588]', ['decode', 1588], ['Q', 4, 3]], ['partial-repair ["decode", 21533]', ['decode', 21533], None], ['control ["decode", 27094]', ['decode', 27094], ['L', 7, 2]], ['control ["encode", ["X", 1]]', ['encode', ['X', 1]], None], ['control ["encode", ["L", 8]]', ['encode', ['L', 8]], None], ['control ["decode", 0]', ['decode', 0], None], ['control ["decode", 32767]', ['decode', 32767], None]]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
regression ["decode", 32647]None['M', 6, 3]Failed
regression ["decode", 1588]None['Q', 4, 3]Failed
partial-repair ["decode", 21533]NoneNonePassed
control ["encode", ["L", 0]]3066030660Passed
control ["encode", ["L", 3]]3087730877Passed
control ["encode", ["L", 5]]2536825368Passed
control ["encode", ["L", 7]]2699826998Passed
control ["encode", ["M", 0]]2152221522Passed

SHA-256 / 5d908b1b5d7a27a5dba1b5102d942fbf52ce4120f03d577c6a552ec24baf67d4

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(op, x):
    EC = {'L': 1, 'M': 0, 'Q': 3, 'H': 2}
    def enc(level, mask):
        data = (EC[level] << 3) | mask
        rem = data << 10
        for i in range(14, 9, -1):
            if rem & (1 << i):
                rem ^= 0x537 << (i - 10)
        return ((data << 10) | rem) ^ 0x5412
    if op == 'encode':
        level, mask = x
        if level not in EC or not 0 <= mask <= 7:
            return None
        return enc(level, mask)
    best = None
    for level in 'LMQH':
        for mask in range(8):
            dist = bin(enc(level, mask) ^ x).count('1')
            if dist <= 4 and (best is None or dist < best[2]):
                best = [level, mask, dist]
    return best
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression ["decode", 32647]', ['decode', 32647], ['M', 6, 3]], ['regression ["decode", 1588]', ['decode', 1588], ['Q', 4, 3]], ['partial-repair ["decode", 21533]', ['decode', 21533], None], ['control ["encode", ["L", 0]]', ['encode', ['L', 0]], 30660], ['control ["encode", ["L", 3]]', ['encode', ['L', 3]], 30877], ['control ["encode", ["L", 5]]', ['encode', ['L', 5]], 25368], ['control ["encode", ["L", 7]]', ['encode', ['L', 7]], 26998], ['control ["encode", ["M", 0]]', ['encode', ['M', 0]], 21522]], [['regression ["decode", 31121]', ['decode', 31121], ['L', 3, 3]], ['regression ["decode", 2309]', ['decode', 2309], ['H', 6, 3]], ['partial-repair ["decode", 21533]', ['decode', 21533], None], ['control ["encode", ["M", 3]]', ['encode', ['M', 3]], 23371], ['control ["encode", ["M", 5]]', ['encode', ['M', 5]], 16590], ['control ["encode", ["M", 7]]', ['encode', ['M', 7]], 19104], ['control ["encode", ["Q", 0]]', ['encode', ['Q', 0]], 13663], ['control ["encode", ["Q", 3]]', ['encode', ['Q', 3]], 14854]], [['regression ["decode", 32647]', ['decode', 32647], ['M', 6, 3]], ['regression ["decode", 1588]', ['decode', 1588], ['Q', 4, 3]], ['partial-repair ["decode", 21533]', ['decode', 21533], None], ['control ["encode", ["Q", 5]]', ['encode', ['Q', 5]], 8579], ['control ["encode", ["Q", 7]]', ['encode', ['Q', 7]], 11245], ['control ["encode", ["H", 0]]', ['encode', ['H', 0]], 5769], ['control ["encode", ["H", 3]]', ['encode', ['H', 3]], 6608], ['control ["encode", ["H", 5]]', ['encode', ['H', 5]], 597]], [['regression ["decode", 31121]', ['decode', 31121], ['L', 3, 3]], ['regression ["decode", 2309]', ['decode', 2309], ['H', 6, 3]], ['partial-repair ["decode", 21533]', ['decode', 21533], None], ['control ["encode", ["H", 7]]', ['encode', ['H', 7]], 2107], ['control ["decode", 19724]', ['decode', 19724], ['H', 6, 1]], ['control ["decode", 6616]', ['decode', 6616], ['H', 3, 1]], ['control ["decode", 30660]', ['decode', 30660], ['L', 0, 0]], ['control ["decode", 2099]', ['decode', 2099], ['H', 7, 1]]], [['regression ["decode", 32647]', ['decode', 32647], ['M', 6, 3]], ['regression ["decode", 1588]', ['decode', 1588], ['Q', 4, 3]], ['partial-repair ["decode", 21533]', ['decode', 21533], None], ['control ["decode", 27094]', ['decode', 27094], ['L', 7, 2]], ['control ["encode", ["X", 1]]', ['encode', ['X', 1]], None], ['control ["encode", ["L", 8]]', ['encode', ['L', 8]], None], ['control ["decode", 0]', ['decode', 0], None], ['control ["decode", 32767]', ['decode', 32767], None]]]
for label, args, expected in fixtures[N - 1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
regression ["decode", 32647]['M', 6, 3]['M', 6, 3]Passed
regression ["decode", 1588]['Q', 4, 3]['Q', 4, 3]Passed
partial-repair ["decode", 21533]['L', 3, 4]NoneFailed
control ["encode", ["L", 0]]3066030660Passed
control ["encode", ["L", 3]]3087730877Passed
control ["encode", ["L", 5]]2536825368Passed
control ["encode", ["L", 7]]2699826998Passed
control ["encode", ["M", 0]]2152221522Passed

SHA-256 / 9805db4c6db181a4673897577bbae6109a91e0c7f91bad4b393f009671312dd7

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

A deterministic, bounded teaching model of the named code under the stated contract; not a production codec. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:48:36.123362+00:00.

Case digest / 4b84683f3450b66914cc6c0e9a0c80c63c65ba1d5dac7a1d3ad37050c121d464