FA-72096 / Error-correcting codes / Open access

Min-sum magnitude includes the edge own message · case 01

The least reliable edge receives its own small magnitude back.

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

ROOT CAUSE

The minimum magnitude is taken over all edges including i.

VERIFIED REPAIR

Take the minimum over the other edges (the second minimum for the minimum edge).

Unsuccessful approach: Always using the second smallest magnitude is wrong for every edge that is not the minimum.

Case contract

Offset min-sum check-node update for LDPC decoding with integer LLRs. For each edge i the outgoing message is the product of the signs of all OTHER incoming LLRs (0 counts as positive) times max(min |other LLR| - beta, 0). Fewer than two edges returns None. Return the outgoing messages in edge order.

Why this case matters

LDPC decoders in Wi-Fi, 5G and SSD controllers run this update billions of times per second.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(llrs, beta):
    if len(llrs) < 2:
        return None
    out = []
    for i in range(len(llrs)):
        others = llrs[:i] + llrs[i + 1:]
        sign = 1
        for v in others:
            if v < 0:
                sign = -sign
        mag = min(abs(v) for v in llrs)
        out.append(sign * max(mag - beta, 0))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[3, 4], 0]', [[3, 4], 0], [4, 3]], ['regression [[12, -11, -9, -2, 8, -3], 1]', [[12, -11, -9, -2, 8, -3], 1], [1, -1, -1, -2, 1, -1]], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-3, 6], 2]', [[-3, 6], 2], [4, -1]], ['control [[3, -8], 3]', [[3, -8], 3], [-5, 0]]], [['regression [[1, -2, -3, -4], 1]', [[1, -2, -3, -4], 1], [-1, 0, 0, 0]], ['regression [[-3, 6], 2]', [[-3, 6], 2], [4, -1]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-3, 6], 3]', [[-3, 6], 3], [3, 0]], ['control [[-5, -9], 0]', [[-5, -9], 0], [-9, -5]]], [['regression [[6, -11, 10, -5, 2, 8], 1]', [[6, -11, 10, -5, 2, 8], 1], [1, -1, 1, -1, 4, 1]], ['regression [[1, 7, -2], 0]', [[1, 7, -2], 0], [-2, -1, 1]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[4, -9, -7, -9, 6], 3]', [[4, -9, -7, -9, 6], 3], [-3, 1, 1, 1, -1]], ['control [[12, 2], 3]', [[12, 2], 3], [0, 9]]], [['regression [[-5, -9], 0]', [[-5, -9], 0], [-9, -5]], ['regression [[-6, 10, 0], 1]', [[-6, 10, 0], 1], [0, 0, -5]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[2, 4], 3]', [[2, 4], 3], [1, 0]], ['control [[1, 11], 2]', [[1, 11], 2], [9, 0]]], [['regression [[5, -12, -9], 3]', [[5, -12, -9], 3], [6, -2, -2]], ['regression [[4, -9, -7, -9, 6], 3]', [[4, -9, -7, -9, 6], 3], [-3, 1, 1, 1, -1]], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[5, -3], 1]', [[5, -3], 1], [-2, 4]], ['control [[0, -4, 6], 1]', [[0, -4, 6], 1], [-3, 0, 0]]]]
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 [[3, 4], 0]	`[3, 3]`	`[4, 3]`	Failed
regression [[12, -11, -9, -2, 8, -3], 1]	`[1, -1, -1, -1, 1, -1]`	`[1, -1, -1, -2, 1, -1]`	Failed
control [[-2, -12, 2, -7, 10, 7], 2]	`[0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0]`	Passed
control [[-12, 7, -1, 8, 11, -2], 3]	`[0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0]`	Passed
control [[2, 2, 2], 3]	`[0, 0, 0]`	`[0, 0, 0]`	Passed
control [[-1], 0]	`None`	`None`	Passed
control [[-3, 6], 2]	`[1, -1]`	`[4, -1]`	Failed
control [[3, -8], 3]	`[0, 0]`	`[-5, 0]`	Failed

SHA-256 / 185e4a68651ee85160f7e7f760adec19891d1377ac2866023ccf69681e2c0b37

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(llrs, beta):
    if len(llrs) < 2:
        return None
    out = []
    for i in range(len(llrs)):
        others = llrs[:i] + llrs[i + 1:]
        sign = 1
        for v in others:
            if v < 0:
                sign = -sign
        mag = sorted(abs(v) for v in llrs)[1]
        out.append(sign * max(mag - beta, 0))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[3, 4], 0]', [[3, 4], 0], [4, 3]], ['regression [[12, -11, -9, -2, 8, -3], 1]', [[12, -11, -9, -2, 8, -3], 1], [1, -1, -1, -2, 1, -1]], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-3, 6], 2]', [[-3, 6], 2], [4, -1]], ['control [[3, -8], 3]', [[3, -8], 3], [-5, 0]]], [['regression [[1, -2, -3, -4], 1]', [[1, -2, -3, -4], 1], [-1, 0, 0, 0]], ['regression [[-3, 6], 2]', [[-3, 6], 2], [4, -1]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-3, 6], 3]', [[-3, 6], 3], [3, 0]], ['control [[-5, -9], 0]', [[-5, -9], 0], [-9, -5]]], [['regression [[6, -11, 10, -5, 2, 8], 1]', [[6, -11, 10, -5, 2, 8], 1], [1, -1, 1, -1, 4, 1]], ['regression [[1, 7, -2], 0]', [[1, 7, -2], 0], [-2, -1, 1]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[4, -9, -7, -9, 6], 3]', [[4, -9, -7, -9, 6], 3], [-3, 1, 1, 1, -1]], ['control [[12, 2], 3]', [[12, 2], 3], [0, 9]]], [['regression [[-5, -9], 0]', [[-5, -9], 0], [-9, -5]], ['regression [[-6, 10, 0], 1]', [[-6, 10, 0], 1], [0, 0, -5]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[2, 4], 3]', [[2, 4], 3], [1, 0]], ['control [[1, 11], 2]', [[1, 11], 2], [9, 0]]], [['regression [[5, -12, -9], 3]', [[5, -12, -9], 3], [6, -2, -2]], ['regression [[4, -9, -7, -9, 6], 3]', [[4, -9, -7, -9, 6], 3], [-3, 1, 1, 1, -1]], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[5, -3], 1]', [[5, -3], 1], [-2, 4]], ['control [[0, -4, 6], 1]', [[0, -4, 6], 1], [-3, 0, 0]]]]
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 [[3, 4], 0]	`[4, 4]`	`[4, 3]`	Failed
regression [[12, -11, -9, -2, 8, -3], 1]	`[2, -2, -2, -2, 2, -2]`	`[1, -1, -1, -2, 1, -1]`	Failed
control [[-2, -12, 2, -7, 10, 7], 2]	`[0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0]`	Passed
control [[-12, 7, -1, 8, 11, -2], 3]	`[0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0]`	Passed
control [[2, 2, 2], 3]	`[0, 0, 0]`	`[0, 0, 0]`	Passed
control [[-1], 0]	`None`	`None`	Passed
control [[-3, 6], 2]	`[4, -4]`	`[4, -1]`	Failed
control [[3, -8], 3]	`[-5, 5]`	`[-5, 0]`	Failed

SHA-256 / 7d47765a4a51ae1bf6c30a1991e3575a1c11b5bb6f6a3c879d87907db9660e49

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(llrs, beta):
    if len(llrs) < 2:
        return None
    out = []
    for i in range(len(llrs)):
        others = llrs[:i] + llrs[i + 1:]
        sign = 1
        for v in others:
            if v < 0:
                sign = -sign
        mag = min(abs(v) for v in others)
        out.append(sign * max(mag - beta, 0))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[3, 4], 0]', [[3, 4], 0], [4, 3]], ['regression [[12, -11, -9, -2, 8, -3], 1]', [[12, -11, -9, -2, 8, -3], 1], [1, -1, -1, -2, 1, -1]], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-3, 6], 2]', [[-3, 6], 2], [4, -1]], ['control [[3, -8], 3]', [[3, -8], 3], [-5, 0]]], [['regression [[1, -2, -3, -4], 1]', [[1, -2, -3, -4], 1], [-1, 0, 0, 0]], ['regression [[-3, 6], 2]', [[-3, 6], 2], [4, -1]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-3, 6], 3]', [[-3, 6], 3], [3, 0]], ['control [[-5, -9], 0]', [[-5, -9], 0], [-9, -5]]], [['regression [[6, -11, 10, -5, 2, 8], 1]', [[6, -11, 10, -5, 2, 8], 1], [1, -1, 1, -1, 4, 1]], ['regression [[1, 7, -2], 0]', [[1, 7, -2], 0], [-2, -1, 1]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[4, -9, -7, -9, 6], 3]', [[4, -9, -7, -9, 6], 3], [-3, 1, 1, 1, -1]], ['control [[12, 2], 3]', [[12, 2], 3], [0, 9]]], [['regression [[-5, -9], 0]', [[-5, -9], 0], [-9, -5]], ['regression [[-6, 10, 0], 1]', [[-6, 10, 0], 1], [0, 0, -5]], ['control [[-1], 0]', [[-1], 0], None], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[2, 4], 3]', [[2, 4], 3], [1, 0]], ['control [[1, 11], 2]', [[1, 11], 2], [9, 0]]], [['regression [[5, -12, -9], 3]', [[5, -12, -9], 3], [6, -2, -2]], ['regression [[4, -9, -7, -9, 6], 3]', [[4, -9, -7, -9, 6], 3], [-3, 1, 1, 1, -1]], ['control [[-2, -12, 2, -7, 10, 7], 2]', [[-2, -12, 2, -7, 10, 7], 2], [0, 0, 0, 0, 0, 0]], ['control [[-12, 7, -1, 8, 11, -2], 3]', [[-12, 7, -1, 8, 11, -2], 3], [0, 0, 0, 0, 0, 0]], ['control [[2, 2, 2], 3]', [[2, 2, 2], 3], [0, 0, 0]], ['control [[-1], 0]', [[-1], 0], None], ['control [[5, -3], 1]', [[5, -3], 1], [-2, 4]], ['control [[0, -4, 6], 1]', [[0, -4, 6], 1], [-3, 0, 0]]]]
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 [[3, 4], 0]	`[4, 3]`	`[4, 3]`	Passed
regression [[12, -11, -9, -2, 8, -3], 1]	`[1, -1, -1, -2, 1, -1]`	`[1, -1, -1, -2, 1, -1]`	Passed
control [[-2, -12, 2, -7, 10, 7], 2]	`[0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0]`	Passed
control [[-12, 7, -1, 8, 11, -2], 3]	`[0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0]`	Passed
control [[2, 2, 2], 3]	`[0, 0, 0]`	`[0, 0, 0]`	Passed
control [[-1], 0]	`None`	`None`	Passed
control [[-3, 6], 2]	`[4, -1]`	`[4, -1]`	Passed
control [[3, -8], 3]	`[-5, 0]`	`[-5, 0]`	Passed

SHA-256 / 0a117c9a6750deb1c7f0327e2af34c02517269cfd7cca0e7e1271485f041afff

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 2026-09-29T14:48:35.428070+00:00.

Case digest / cafa2cdd6a3390fead070d3f257a9826b4242de8ee8fe3982c7190da58f8fd75