FA-72166 / Error-correcting codes / Open access

QR format encoder numbers error-correction levels in order · case 01

Every level except H decodes as a different level.

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

ROOT CAUSE

The level bits are taken as L=0, M=1, Q=2, H=3 (alphabetic order) instead of L=01, M=00, Q=11, H=10.

VERIFIED REPAIR

Use the format indicator bits L=1, M=0, Q=3, H=2.

Unsuccessful approach: Swapping L and M but keeping Q=2 and H=3 still mislabels the high levels.

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': 0, 'M': 1, 'Q': 2, 'H': 3}
    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 ["encode", ["L", 0]]', ['encode', ['L', 0]], 30660], ['regression ["encode", ["L", 3]]', ['encode', ['L', 3]], 30877], ['partial-repair ["encode", ["Q", 0]]', ['encode', ['Q', 0]], 13663], ['partial-repair ["encode", ["Q", 3]]', ['encode', ['Q', 3]], 14854], ['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]], [['regression ["encode", ["L", 7]]', ['encode', ['L', 7]], 26998], ['regression ["encode", ["M", 0]]', ['encode', ['M', 0]], 21522], ['partial-repair ["encode", ["Q", 7]]', ['encode', ['Q', 7]], 11245], ['partial-repair ["encode", ["H", 0]]', ['encode', ['H', 0]], 5769], ['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]], [['regression ["encode", ["M", 5]]', ['encode', ['M', 5]], 16590], ['regression ["encode", ["M", 7]]', ['encode', ['M', 7]], 19104], ['partial-repair ["encode", ["H", 5]]', ['encode', ['H', 5]], 597], ['partial-repair ["encode", ["H", 7]]', ['encode', ['H', 7]], 2107], ['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]], [['regression ["encode", ["Q", 3]]', ['encode', ['Q', 3]], 14854], ['regression ["encode", ["Q", 5]]', ['encode', ['Q', 5]], 8579], ['partial-repair ["decode", 19724]', ['decode', 19724], ['H', 6, 1]], ['partial-repair ["decode", 6616]', ['decode', 6616], ['H', 3, 1]], ['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]], [['regression ["encode", ["H", 0]]', ['encode', ['H', 0]], 5769], ['regression ["encode", ["H", 3]]', ['encode', ['H', 3]], 6608], ['partial-repair ["decode", 2099]', ['decode', 2099], ['H', 7, 1]], ['partial-repair ["encode", ["Q", 0]]', ['encode', ['Q', 0]], 13663], ['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 fixture	Actual	Expected	Outcome
regression ["encode", ["L", 0]]	`21522`	`30660`	Failed
regression ["encode", ["L", 3]]	`23371`	`30877`	Failed
partial-repair ["encode", ["Q", 0]]	`5769`	`13663`	Failed
partial-repair ["encode", ["Q", 3]]	`6608`	`14854`	Failed
control ["encode", ["X", 1]]	`None`	`None`	Passed
control ["encode", ["L", 8]]	`None`	`None`	Passed
control ["decode", 0]	`None`	`None`	Passed
control ["decode", 32767]	`None`	`None`	Passed

SHA-256 / eea715eaa46adbc669e1f7503c72403ed95385623af0bc388d2ba25fb0402d33

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': 2, 'H': 3}
    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 ["encode", ["L", 0]]', ['encode', ['L', 0]], 30660], ['regression ["encode", ["L", 3]]', ['encode', ['L', 3]], 30877], ['partial-repair ["encode", ["Q", 0]]', ['encode', ['Q', 0]], 13663], ['partial-repair ["encode", ["Q", 3]]', ['encode', ['Q', 3]], 14854], ['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]], [['regression ["encode", ["L", 7]]', ['encode', ['L', 7]], 26998], ['regression ["encode", ["M", 0]]', ['encode', ['M', 0]], 21522], ['partial-repair ["encode", ["Q", 7]]', ['encode', ['Q', 7]], 11245], ['partial-repair ["encode", ["H", 0]]', ['encode', ['H', 0]], 5769], ['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]], [['regression ["encode", ["M", 5]]', ['encode', ['M', 5]], 16590], ['regression ["encode", ["M", 7]]', ['encode', ['M', 7]], 19104], ['partial-repair ["encode", ["H", 5]]', ['encode', ['H', 5]], 597], ['partial-repair ["encode", ["H", 7]]', ['encode', ['H', 7]], 2107], ['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]], [['regression ["encode", ["Q", 3]]', ['encode', ['Q', 3]], 14854], ['regression ["encode", ["Q", 5]]', ['encode', ['Q', 5]], 8579], ['partial-repair ["decode", 19724]', ['decode', 19724], ['H', 6, 1]], ['partial-repair ["decode", 6616]', ['decode', 6616], ['H', 3, 1]], ['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]], [['regression ["encode", ["H", 0]]', ['encode', ['H', 0]], 5769], ['regression ["encode", ["H", 3]]', ['encode', ['H', 3]], 6608], ['partial-repair ["decode", 2099]', ['decode', 2099], ['H', 7, 1]], ['partial-repair ["encode", ["Q", 0]]', ['encode', ['Q', 0]], 13663], ['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 fixture	Actual	Expected	Outcome
regression ["encode", ["L", 0]]	`30660`	`30660`	Passed
regression ["encode", ["L", 3]]	`30877`	`30877`	Passed
partial-repair ["encode", ["Q", 0]]	`5769`	`13663`	Failed
partial-repair ["encode", ["Q", 3]]	`6608`	`14854`	Failed
control ["encode", ["X", 1]]	`None`	`None`	Passed
control ["encode", ["L", 8]]	`None`	`None`	Passed
control ["decode", 0]	`None`	`None`	Passed
control ["decode", 32767]	`None`	`None`	Passed

SHA-256 / 59a821936029c53593a035165198b9cb0dfb6cc3b483819f358865595306a069

3 / The verified repair

Exit 0

"""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 ["encode", ["L", 0]]', ['encode', ['L', 0]], 30660], ['regression ["encode", ["L", 3]]', ['encode', ['L', 3]], 30877], ['partial-repair ["encode", ["Q", 0]]', ['encode', ['Q', 0]], 13663], ['partial-repair ["encode", ["Q", 3]]', ['encode', ['Q', 3]], 14854], ['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]], [['regression ["encode", ["L", 7]]', ['encode', ['L', 7]], 26998], ['regression ["encode", ["M", 0]]', ['encode', ['M', 0]], 21522], ['partial-repair ["encode", ["Q", 7]]', ['encode', ['Q', 7]], 11245], ['partial-repair ["encode", ["H", 0]]', ['encode', ['H', 0]], 5769], ['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]], [['regression ["encode", ["M", 5]]', ['encode', ['M', 5]], 16590], ['regression ["encode", ["M", 7]]', ['encode', ['M', 7]], 19104], ['partial-repair ["encode", ["H", 5]]', ['encode', ['H', 5]], 597], ['partial-repair ["encode", ["H", 7]]', ['encode', ['H', 7]], 2107], ['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]], [['regression ["encode", ["Q", 3]]', ['encode', ['Q', 3]], 14854], ['regression ["encode", ["Q", 5]]', ['encode', ['Q', 5]], 8579], ['partial-repair ["decode", 19724]', ['decode', 19724], ['H', 6, 1]], ['partial-repair ["decode", 6616]', ['decode', 6616], ['H', 3, 1]], ['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]], [['regression ["encode", ["H", 0]]', ['encode', ['H', 0]], 5769], ['regression ["encode", ["H", 3]]', ['encode', ['H', 3]], 6608], ['partial-repair ["decode", 2099]', ['decode', 2099], ['H', 7, 1]], ['partial-repair ["encode", ["Q", 0]]', ['encode', ['Q', 0]], 13663], ['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 fixture	Actual	Expected	Outcome
regression ["encode", ["L", 0]]	`30660`	`30660`	Passed
regression ["encode", ["L", 3]]	`30877`	`30877`	Passed
partial-repair ["encode", ["Q", 0]]	`13663`	`13663`	Passed
partial-repair ["encode", ["Q", 3]]	`14854`	`14854`	Passed
control ["encode", ["X", 1]]	`None`	`None`	Passed
control ["encode", ["L", 8]]	`None`	`None`	Passed
control ["decode", 0]	`None`	`None`	Passed
control ["decode", 32767]	`None`	`None`	Passed

SHA-256 / 829296a4149dc31ca9887e06c12caeb0f3dd16eb65e15cf2c9b6830b4359d653

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 2026-09-29T14:48:36.039653+00:00.

Case digest / e8861cc0d220bb9be3793b672883bb520355b02b3cb449c6484d1765851fd617