FA-71866 / Error-correcting codes / Open access

Hamming encoder writes the nibble in reverse order · case 01

Decoders recover the data bits reversed.

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

ROOT CAUSE

The data bits are placed as d4, d3, d2, d1 across positions 3, 5, 6, 7.

VERIFIED REPAIR

Place d1 at position 3, d2 at 5, d3 at 6 and d4 at 7.

Unsuccessful approach: Swapping only d2 and d3 still misplaces half the data.

Case contract

Hamming(7,4) encoder: data d1..d4 go to positions 3, 5, 6, 7; the parity bit at position p (1, 2, 4) is the XOR of all other positions whose index has bit p set. Return the 7-bit codeword (position 1 first).

Why this case matters

Low-cost telemetry links protect nibbles with Hamming(7,4).

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(d):
    d1, d2, d3, d4 = d
    c = [0, 0, d4, 0, d3, d2, d1]
    for p in (1, 2, 4):
        v = 0
        for pos in range(1, 8):
            if pos & p and pos != p:
                v ^= c[pos - 1]
        c[p - 1] = v
    return c
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[0, 0, 0, 1]]', [[0, 0, 0, 1]], [1, 1, 0, 1, 0, 0, 1]], ['regression [[0, 0, 1, 0]]', [[0, 0, 1, 0]], [0, 1, 0, 1, 0, 1, 0]], ['partial-repair [[0, 0, 1, 1]]', [[0, 0, 1, 1]], [1, 0, 0, 0, 0, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 1, 0, 0]]', [[0, 1, 0, 0]], [1, 0, 0, 1, 1, 0, 0]]], [['regression [[0, 1, 0, 0]]', [[0, 1, 0, 0]], [1, 0, 0, 1, 1, 0, 0]], ['regression [[0, 1, 0, 1]]', [[0, 1, 0, 1]], [0, 1, 0, 0, 1, 0, 1]], ['partial-repair [[1, 0, 1, 0]]', [[1, 0, 1, 0]], [1, 0, 1, 1, 0, 1, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[1, 0, 1, 1]]', [[1, 0, 1, 1]], [0, 1, 1, 0, 0, 1, 1]]], [['regression [[1, 0, 0, 0]]', [[1, 0, 0, 0]], [1, 1, 1, 0, 0, 0, 0]], ['regression [[1, 0, 1, 0]]', [[1, 0, 1, 0]], [1, 0, 1, 1, 0, 1, 0]], ['partial-repair [[1, 1, 0, 0]]', [[1, 1, 0, 0]], [0, 1, 1, 1, 1, 0, 0]], ['partial-repair [[1, 1, 0, 1]]', [[1, 1, 0, 1]], [1, 0, 1, 0, 1, 0, 1]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]]], [['regression [[1, 1, 0, 0]]', [[1, 1, 0, 0]], [0, 1, 1, 1, 1, 0, 0]], ['regression [[1, 1, 0, 1]]', [[1, 1, 0, 1]], [1, 0, 1, 0, 1, 0, 1]], ['partial-repair [[0, 0, 1, 1]]', [[0, 0, 1, 1]], [1, 0, 0, 0, 0, 1, 1]], ['partial-repair [[0, 1, 0, 0]]', [[0, 1, 0, 0]], [1, 0, 0, 1, 1, 0, 0]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]]], [['regression [[0, 0, 0, 1]]', [[0, 0, 0, 1]], [1, 1, 0, 1, 0, 0, 1]], ['regression [[0, 0, 1, 0]]', [[0, 0, 1, 0]], [0, 1, 0, 1, 0, 1, 0]], ['partial-repair [[1, 0, 1, 0]]', [[1, 0, 1, 0]], [1, 0, 1, 1, 0, 1, 0]], ['partial-repair [[1, 0, 1, 1]]', [[1, 0, 1, 1]], [0, 1, 1, 0, 0, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]]]]
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 [[0, 0, 0, 1]]	`[1, 1, 1, 0, 0, 0, 0]`	`[1, 1, 0, 1, 0, 0, 1]`	Failed
regression [[0, 0, 1, 0]]	`[1, 0, 0, 1, 1, 0, 0]`	`[0, 1, 0, 1, 0, 1, 0]`	Failed
partial-repair [[0, 0, 1, 1]]	`[0, 1, 1, 1, 1, 0, 0]`	`[1, 0, 0, 0, 0, 1, 1]`	Failed
control [[0, 0, 0, 0]]	`[0, 0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0, 0]`	Passed
control [[0, 1, 1, 0]]	`[1, 1, 0, 0, 1, 1, 0]`	`[1, 1, 0, 0, 1, 1, 0]`	Passed
control [[1, 0, 0, 1]]	`[0, 0, 1, 1, 0, 0, 1]`	`[0, 0, 1, 1, 0, 0, 1]`	Passed
control [[1, 1, 1, 1]]	`[1, 1, 1, 1, 1, 1, 1]`	`[1, 1, 1, 1, 1, 1, 1]`	Passed
control [[0, 1, 0, 0]]	`[0, 1, 0, 1, 0, 1, 0]`	`[1, 0, 0, 1, 1, 0, 0]`	Failed

SHA-256 / 0a23c7200eab4c824d9112af3306717383d06fcf6b42d64adcd4cf8e8cd75cce

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(d):
    d1, d2, d3, d4 = d
    c = [0, 0, d1, 0, d3, d2, d4]
    for p in (1, 2, 4):
        v = 0
        for pos in range(1, 8):
            if pos & p and pos != p:
                v ^= c[pos - 1]
        c[p - 1] = v
    return c
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[0, 0, 0, 1]]', [[0, 0, 0, 1]], [1, 1, 0, 1, 0, 0, 1]], ['regression [[0, 0, 1, 0]]', [[0, 0, 1, 0]], [0, 1, 0, 1, 0, 1, 0]], ['partial-repair [[0, 0, 1, 1]]', [[0, 0, 1, 1]], [1, 0, 0, 0, 0, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 1, 0, 0]]', [[0, 1, 0, 0]], [1, 0, 0, 1, 1, 0, 0]]], [['regression [[0, 1, 0, 0]]', [[0, 1, 0, 0]], [1, 0, 0, 1, 1, 0, 0]], ['regression [[0, 1, 0, 1]]', [[0, 1, 0, 1]], [0, 1, 0, 0, 1, 0, 1]], ['partial-repair [[1, 0, 1, 0]]', [[1, 0, 1, 0]], [1, 0, 1, 1, 0, 1, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[1, 0, 1, 1]]', [[1, 0, 1, 1]], [0, 1, 1, 0, 0, 1, 1]]], [['regression [[1, 0, 0, 0]]', [[1, 0, 0, 0]], [1, 1, 1, 0, 0, 0, 0]], ['regression [[1, 0, 1, 0]]', [[1, 0, 1, 0]], [1, 0, 1, 1, 0, 1, 0]], ['partial-repair [[1, 1, 0, 0]]', [[1, 1, 0, 0]], [0, 1, 1, 1, 1, 0, 0]], ['partial-repair [[1, 1, 0, 1]]', [[1, 1, 0, 1]], [1, 0, 1, 0, 1, 0, 1]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]]], [['regression [[1, 1, 0, 0]]', [[1, 1, 0, 0]], [0, 1, 1, 1, 1, 0, 0]], ['regression [[1, 1, 0, 1]]', [[1, 1, 0, 1]], [1, 0, 1, 0, 1, 0, 1]], ['partial-repair [[0, 0, 1, 1]]', [[0, 0, 1, 1]], [1, 0, 0, 0, 0, 1, 1]], ['partial-repair [[0, 1, 0, 0]]', [[0, 1, 0, 0]], [1, 0, 0, 1, 1, 0, 0]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]]], [['regression [[0, 0, 0, 1]]', [[0, 0, 0, 1]], [1, 1, 0, 1, 0, 0, 1]], ['regression [[0, 0, 1, 0]]', [[0, 0, 1, 0]], [0, 1, 0, 1, 0, 1, 0]], ['partial-repair [[1, 0, 1, 0]]', [[1, 0, 1, 0]], [1, 0, 1, 1, 0, 1, 0]], ['partial-repair [[1, 0, 1, 1]]', [[1, 0, 1, 1]], [0, 1, 1, 0, 0, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]]]]
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 [[0, 0, 0, 1]]	`[1, 1, 0, 1, 0, 0, 1]`	`[1, 1, 0, 1, 0, 0, 1]`	Passed
regression [[0, 0, 1, 0]]	`[1, 0, 0, 1, 1, 0, 0]`	`[0, 1, 0, 1, 0, 1, 0]`	Failed
partial-repair [[0, 0, 1, 1]]	`[0, 1, 0, 0, 1, 0, 1]`	`[1, 0, 0, 0, 0, 1, 1]`	Failed
control [[0, 0, 0, 0]]	`[0, 0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0, 0]`	Passed
control [[0, 1, 1, 0]]	`[1, 1, 0, 0, 1, 1, 0]`	`[1, 1, 0, 0, 1, 1, 0]`	Passed
control [[1, 0, 0, 1]]	`[0, 0, 1, 1, 0, 0, 1]`	`[0, 0, 1, 1, 0, 0, 1]`	Passed
control [[1, 1, 1, 1]]	`[1, 1, 1, 1, 1, 1, 1]`	`[1, 1, 1, 1, 1, 1, 1]`	Passed
control [[0, 1, 0, 0]]	`[0, 1, 0, 1, 0, 1, 0]`	`[1, 0, 0, 1, 1, 0, 0]`	Failed

SHA-256 / 8b8f08a1f15402fa326ecfc0c3864fdcdf7f1d002da8415f287e841010f4bb64

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(d):
    d1, d2, d3, d4 = d
    c = [0, 0, d1, 0, d2, d3, d4]
    for p in (1, 2, 4):
        v = 0
        for pos in range(1, 8):
            if pos & p and pos != p:
                v ^= c[pos - 1]
        c[p - 1] = v
    return c
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [[0, 0, 0, 1]]', [[0, 0, 0, 1]], [1, 1, 0, 1, 0, 0, 1]], ['regression [[0, 0, 1, 0]]', [[0, 0, 1, 0]], [0, 1, 0, 1, 0, 1, 0]], ['partial-repair [[0, 0, 1, 1]]', [[0, 0, 1, 1]], [1, 0, 0, 0, 0, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 1, 0, 0]]', [[0, 1, 0, 0]], [1, 0, 0, 1, 1, 0, 0]]], [['regression [[0, 1, 0, 0]]', [[0, 1, 0, 0]], [1, 0, 0, 1, 1, 0, 0]], ['regression [[0, 1, 0, 1]]', [[0, 1, 0, 1]], [0, 1, 0, 0, 1, 0, 1]], ['partial-repair [[1, 0, 1, 0]]', [[1, 0, 1, 0]], [1, 0, 1, 1, 0, 1, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[1, 0, 1, 1]]', [[1, 0, 1, 1]], [0, 1, 1, 0, 0, 1, 1]]], [['regression [[1, 0, 0, 0]]', [[1, 0, 0, 0]], [1, 1, 1, 0, 0, 0, 0]], ['regression [[1, 0, 1, 0]]', [[1, 0, 1, 0]], [1, 0, 1, 1, 0, 1, 0]], ['partial-repair [[1, 1, 0, 0]]', [[1, 1, 0, 0]], [0, 1, 1, 1, 1, 0, 0]], ['partial-repair [[1, 1, 0, 1]]', [[1, 1, 0, 1]], [1, 0, 1, 0, 1, 0, 1]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]]], [['regression [[1, 1, 0, 0]]', [[1, 1, 0, 0]], [0, 1, 1, 1, 1, 0, 0]], ['regression [[1, 1, 0, 1]]', [[1, 1, 0, 1]], [1, 0, 1, 0, 1, 0, 1]], ['partial-repair [[0, 0, 1, 1]]', [[0, 0, 1, 1]], [1, 0, 0, 0, 0, 1, 1]], ['partial-repair [[0, 1, 0, 0]]', [[0, 1, 0, 0]], [1, 0, 0, 1, 1, 0, 0]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]]], [['regression [[0, 0, 0, 1]]', [[0, 0, 0, 1]], [1, 1, 0, 1, 0, 0, 1]], ['regression [[0, 0, 1, 0]]', [[0, 0, 1, 0]], [0, 1, 0, 1, 0, 1, 0]], ['partial-repair [[1, 0, 1, 0]]', [[1, 0, 1, 0]], [1, 0, 1, 1, 0, 1, 0]], ['partial-repair [[1, 0, 1, 1]]', [[1, 0, 1, 1]], [0, 1, 1, 0, 0, 1, 1]], ['control [[0, 0, 0, 0]]', [[0, 0, 0, 0]], [0, 0, 0, 0, 0, 0, 0]], ['control [[0, 1, 1, 0]]', [[0, 1, 1, 0]], [1, 1, 0, 0, 1, 1, 0]], ['control [[1, 0, 0, 1]]', [[1, 0, 0, 1]], [0, 0, 1, 1, 0, 0, 1]], ['control [[1, 1, 1, 1]]', [[1, 1, 1, 1]], [1, 1, 1, 1, 1, 1, 1]]]]
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 [[0, 0, 0, 1]]	`[1, 1, 0, 1, 0, 0, 1]`	`[1, 1, 0, 1, 0, 0, 1]`	Passed
regression [[0, 0, 1, 0]]	`[0, 1, 0, 1, 0, 1, 0]`	`[0, 1, 0, 1, 0, 1, 0]`	Passed
partial-repair [[0, 0, 1, 1]]	`[1, 0, 0, 0, 0, 1, 1]`	`[1, 0, 0, 0, 0, 1, 1]`	Passed
control [[0, 0, 0, 0]]	`[0, 0, 0, 0, 0, 0, 0]`	`[0, 0, 0, 0, 0, 0, 0]`	Passed
control [[0, 1, 1, 0]]	`[1, 1, 0, 0, 1, 1, 0]`	`[1, 1, 0, 0, 1, 1, 0]`	Passed
control [[1, 0, 0, 1]]	`[0, 0, 1, 1, 0, 0, 1]`	`[0, 0, 1, 1, 0, 0, 1]`	Passed
control [[1, 1, 1, 1]]	`[1, 1, 1, 1, 1, 1, 1]`	`[1, 1, 1, 1, 1, 1, 1]`	Passed
control [[0, 1, 0, 0]]	`[1, 0, 0, 1, 1, 0, 0]`	`[1, 0, 0, 1, 1, 0, 0]`	Passed

SHA-256 / 07fd1cd37bcafaf27500981719d5cfc1f054375be8735795cd1aa9769722c2d3

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 2026-09-29T14:48:33.463074+00:00.

Case digest / da70fb6fc5116f8cadab32f38a7a436efae20db299b896c0ebce5b2ca5b4cdf6