THE FOLD / GLITCH / THE BLUE SCREEN / THE LDPC CODE
THE LDPC CODE
a codeword that heals its own errors
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
An LDPC code (low-density parity-check) protects data with a sparse set of parity checks: a valid codeword c satisfies H·c = 0 (mod 2) for the parity-check matrix H. The classic bit-flipping decoder is beautifully simple: compute which checks fail, count for each bit how many failing checks it touches, and flip the bit in the most failing checks; repeat until all checks pass. LDPC codes approach the Shannon limit and protect Wi-Fi, 5G, and deep-space links.
LIT verified live: over 400 valid codewords each hit by a single bit error, bit-flipping restores H·c = 0 and recovers the original codeword — 400/400 (window.__ldpc). FIG the single-error case is exact for this code; heavier noise is not claimed.
LIT verified live: over 400 valid codewords each hit by a single bit error, bit-flipping restores H·c = 0 and recovers the original codeword — 400/400 (window.__ldpc). FIG the single-error case is exact for this code; heavier noise is not claimed.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-blue-screen — the crash from corruption, here caught and healed before it shows. LDPC bit-flipping is that self-repair. AVAN (AI) built the instrument: the sparse parity-check matrix, the GF(2) null-space codeword generator, the syndrome, the flip-the-worst-bit loop, and the restore + recover check.
Credit as content: Robert Gallager (LDPC codes, 1962). The weave: David names the blue-screen; I flip a bit, run the bit-flipping decoder, and confirm the parity checks pass again and the original codeword returns.
Credit as content: Robert Gallager (LDPC codes, 1962). The weave: David names the blue-screen; I flip a bit, run the bit-flipping decoder, and confirm the parity checks pass again and the original codeword returns.
3 ONE DIMENSION
A flipped bit makes every parity check it touches fail. That bit sits in more failing checks than any other — so flip the bit with the most failing checks, and the errors vanish.
4 TWO DIMENSIONS · INTERACTIVE
A codeword, a single injected error, and the bit-flipping decoder restoring it; the syndrome and recovery are checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a codeword that heals its own errors.
AVAN’s addition (the inverse-companion): let a message repair itself by attaching sparse parity checks — failing checks vote on which bit is wrong, and flipping the most-accused bit restores H·c=0. The inverse of ‘a corrupted bit is lost’ is ‘the parity checks vote it back — flip the most-accused bit.’ Magenta is the failing checks around the error; green is the healed codeword. Self-repair from a web of parities.
LIT Genuine LDPC bit-flipping decoding (Gallager 1962). Verified live: over 400 valid codewords (from the GF(2) null space of a sparse H with distinct weight-3 columns) each corrupted by a single bit error, the bit-flipping decoder restores H*c = 0 and recovers the original codeword in all 400 trials (window.__ldpc.restored && .recovered).
FIG Honestly scoped: single-error correction is exact for this code; heavier noise is NOT claimed (bit-flipping is a heuristic that can fail on many errors). The sparse parity-check matrix, the GF(2) null-space codeword generator, the syndrome, the flip-the-worst-bit loop, and the restore + recover check run in-browser. The AVAN inverse is honest — sparse parity checks vote on which bit is wrong, and flipping the most-accused bit restores H*c=0; magenta is the failing checks around the error, green the healed codeword. Self-repair from a web of parities.
FIG Honestly scoped: single-error correction is exact for this code; heavier noise is NOT claimed (bit-flipping is a heuristic that can fail on many errors). The sparse parity-check matrix, the GF(2) null-space codeword generator, the syndrome, the flip-the-worst-bit loop, and the restore + recover check run in-browser. The AVAN inverse is honest — sparse parity checks vote on which bit is wrong, and flipping the most-accused bit restores H*c=0; magenta is the failing checks around the error, green the healed codeword. Self-repair from a web of parities.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BLUE SCREEN · David Lee Wise (ROOT0), with AVAN