FA-72261 / Error-correcting codes / Open access

Burst capability ignores symbol boundary straddling · case 01

Quoted burst capability is too large by up to one symbol.

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

ROOT CAUSE

The capability is tDm, assuming bursts start on symbol boundaries.

THE FAILURE

The capability is t*D*m, assuming bursts start on symbol boundaries.

Unsuccessful approach: Subtracting a whole symbol without adding back the single bit understates the capability by one.

Case contract

Guaranteed burst correction of an RS(n, k) code with m-bit symbols interleaved to depth D. Each codeword corrects t = floor((n - k) / 2) symbol errors; D interleaved codewords correct any burst of at most (t*D - 1)*m + 1 bits (a burst that straddles symbol boundaries touches one extra symbol). t = 0 gives 0. Invalid parameters return None.

Why this case matters

Optical, DVB and storage link budgets quote the longest correctable burst for an interleaved RS code.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(n, k, m, depth):
    if not 0 < k < n or m < 1 or depth < 1:
        return None
    t = (n - k) // 2
    burst = t * depth * m if t else 0
    return [t, burst]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [255, 223, 8, 1]', [255, 223, 8, 1], [16, 121]], ['regression [255, 223, 8, 4]', [255, 223, 8, 4], [16, 505]], ['control [255, 254, 8, 5]', [255, 254, 8, 5], [0, 0]], ['control [5, 5, 8, 1]', [5, 5, 8, 1], None], ['control [5, 2, 0, 1]', [5, 2, 0, 1], None], ['control [7, 6, 1, 1]', [7, 6, 1, 1], [0, 0]], ['control [15, 11, 4, 3]', [15, 11, 4, 3], [2, 21]], ['control [7, 3, 3, 2]', [7, 3, 3, 2], [2, 10]]], [['regression [204, 188, 8, 12]', [204, 188, 8, 12], [8, 761]], ['regression [15, 11, 4, 3]', [15, 11, 4, 3], [2, 21]], ['control [5, 5, 8, 1]', [5, 5, 8, 1], None], ['control [5, 2, 0, 1]', [5, 2, 0, 1], None], ['control [7, 6, 1, 1]', [7, 6, 1, 1], [0, 0]], ['control [255, 254, 8, 5]', [255, 254, 8, 5], [0, 0]], ['control [10, 7, 2, 3]', [10, 7, 2, 3], [1, 5]], ['control [255, 253, 8, 1]', [255, 253, 8, 1], [1, 1]]], [['regression [31, 25, 5, 1]', [31, 25, 5, 1], [3, 11]], ['regression [15, 9, 4, 1]', [15, 9, 4, 1], [3, 9]], ['control [5, 2, 0, 1]', [5, 2, 0, 1], None], ['control [7, 6, 1, 1]', [7, 6, 1, 1], [0, 0]], ['control [255, 254, 8, 5]', [255, 254, 8, 5], [0, 0]], ['control [5, 5, 8, 1]', [5, 5, 8, 1], None], ['control [255, 223, 8, 1]', [255, 223, 8, 1], [16, 121]], ['control [255, 223, 8, 4]', [255, 223, 8, 4], [16, 505]]], [['regression [255, 253, 8, 1]', [255, 253, 8, 1], [1, 1]], ['regression [7, 5, 3, 4]', [7, 5, 3, 4], [1, 10]], ['control [7, 6, 1, 1]', [7, 6, 1, 1], [0, 0]], ['control [255, 254, 8, 5]', [255, 254, 8, 5], [0, 0]], ['control [5, 5, 8, 1]', [5, 5, 8, 1], None], ['control [5, 2, 0, 1]', [5, 2, 0, 1], None], ['control [204, 188, 8, 12]', [204, 188, 8, 12], [8, 761]], ['control [15, 11, 4, 3]', [15, 11, 4, 3], [2, 21]]], [['regression [255, 223, 8, 4]', [255, 223, 8, 4], [16, 505]], ['regression [255, 239, 8, 2]', [255, 239, 8, 2], [8, 121]], ['partial-repair [255, 223, 8, 1]', [255, 223, 8, 1], [16, 121]], ['control [255, 254, 8, 5]', [255, 254, 8, 5], [0, 0]], ['control [5, 5, 8, 1]', [5, 5, 8, 1], None], ['control [5, 2, 0, 1]', [5, 2, 0, 1], None], ['control [7, 6, 1, 1]', [7, 6, 1, 1], [0, 0]], ['control [15, 9, 4, 1]', [15, 9, 4, 1], [3, 9]]]]
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 [255, 223, 8, 1]	`[16, 128]`	`[16, 121]`	Failed
regression [255, 223, 8, 4]	`[16, 512]`	`[16, 505]`	Failed
control [255, 254, 8, 5]	`[0, 0]`	`[0, 0]`	Passed
control [5, 5, 8, 1]	`None`	`None`	Passed
control [5, 2, 0, 1]	`None`	`None`	Passed
control [7, 6, 1, 1]	`[0, 0]`	`[0, 0]`	Passed
control [15, 11, 4, 3]	`[2, 24]`	`[2, 21]`	Failed
control [7, 3, 3, 2]	`[2, 12]`	`[2, 10]`	Failed

SHA-256 / bb0ad85c77a0ddc897b2006efb4293fa28c74d6b0bb5d3974e41f7c574871f06

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(n, k, m, depth):
    if not 0 < k < n or m < 1 or depth < 1:
        return None
    t = (n - k) // 2
    burst = t * depth * m - m if t else 0
    return [t, burst]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [255, 223, 8, 1]', [255, 223, 8, 1], [16, 121]], ['regression [255, 223, 8, 4]', [255, 223, 8, 4], [16, 505]], ['control [255, 254, 8, 5]', [255, 254, 8, 5], [0, 0]], ['control [5, 5, 8, 1]', [5, 5, 8, 1], None], ['control [5, 2, 0, 1]', [5, 2, 0, 1], None], ['control [7, 6, 1, 1]', [7, 6, 1, 1], [0, 0]], ['control [15, 11, 4, 3]', [15, 11, 4, 3], [2, 21]], ['control [7, 3, 3, 2]', [7, 3, 3, 2], [2, 10]]], [['regression [204, 188, 8, 12]', [204, 188, 8, 12], [8, 761]], ['regression [15, 11, 4, 3]', [15, 11, 4, 3], [2, 21]], ['control [5, 5, 8, 1]', [5, 5, 8, 1], None], ['control [5, 2, 0, 1]', [5, 2, 0, 1], None], ['control [7, 6, 1, 1]', [7, 6, 1, 1], [0, 0]], ['control [255, 254, 8, 5]', [255, 254, 8, 5], [0, 0]], ['control [10, 7, 2, 3]', [10, 7, 2, 3], [1, 5]], ['control [255, 253, 8, 1]', [255, 253, 8, 1], [1, 1]]], [['regression [31, 25, 5, 1]', [31, 25, 5, 1], [3, 11]], ['regression [15, 9, 4, 1]', [15, 9, 4, 1], [3, 9]], ['control [5, 2, 0, 1]', [5, 2, 0, 1], None], ['control [7, 6, 1, 1]', [7, 6, 1, 1], [0, 0]], ['control [255, 254, 8, 5]', [255, 254, 8, 5], [0, 0]], ['control [5, 5, 8, 1]', [5, 5, 8, 1], None], ['control [255, 223, 8, 1]', [255, 223, 8, 1], [16, 121]], ['control [255, 223, 8, 4]', [255, 223, 8, 4], [16, 505]]], [['regression [255, 253, 8, 1]', [255, 253, 8, 1], [1, 1]], ['regression [7, 5, 3, 4]', [7, 5, 3, 4], [1, 10]], ['control [7, 6, 1, 1]', [7, 6, 1, 1], [0, 0]], ['control [255, 254, 8, 5]', [255, 254, 8, 5], [0, 0]], ['control [5, 5, 8, 1]', [5, 5, 8, 1], None], ['control [5, 2, 0, 1]', [5, 2, 0, 1], None], ['control [204, 188, 8, 12]', [204, 188, 8, 12], [8, 761]], ['control [15, 11, 4, 3]', [15, 11, 4, 3], [2, 21]]], [['regression [255, 223, 8, 4]', [255, 223, 8, 4], [16, 505]], ['regression [255, 239, 8, 2]', [255, 239, 8, 2], [8, 121]], ['partial-repair [255, 223, 8, 1]', [255, 223, 8, 1], [16, 121]], ['control [255, 254, 8, 5]', [255, 254, 8, 5], [0, 0]], ['control [5, 5, 8, 1]', [5, 5, 8, 1], None], ['control [5, 2, 0, 1]', [5, 2, 0, 1], None], ['control [7, 6, 1, 1]', [7, 6, 1, 1], [0, 0]], ['control [15, 9, 4, 1]', [15, 9, 4, 1], [3, 9]]]]
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 [255, 223, 8, 1]	`[16, 120]`	`[16, 121]`	Failed
regression [255, 223, 8, 4]	`[16, 504]`	`[16, 505]`	Failed
control [255, 254, 8, 5]	`[0, 0]`	`[0, 0]`	Passed
control [5, 5, 8, 1]	`None`	`None`	Passed
control [5, 2, 0, 1]	`None`	`None`	Passed
control [7, 6, 1, 1]	`[0, 0]`	`[0, 0]`	Passed
control [15, 11, 4, 3]	`[2, 20]`	`[2, 21]`	Failed
control [7, 3, 3, 2]	`[2, 9]`	`[2, 10]`	Failed

SHA-256 / 7c13cd0af7258c5e46765c8a92bc898f1bba575249c28be861c76502cdaf90a1

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.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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 2026-09-29T14:48:36.943114+00:00.

Case digest / bc389daa3622c1ee0d601e19f42abcc040e6af79dc7e9a46ef14508df1f01d16