THE FOLD / GLITCH / SEGFAULT / THE REED-MULLER
THE REED-MULLER
a code folded from itself
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Reed–Muller code RM(r, m) is an error-correcting code built from low-degree Boolean polynomials: its codewords are the truth-tables of every multilinear polynomial of degree ≤ r in m variables. That gives length 2m, dimension ∑i≤r C(m, i), and a clean minimum distance of 2m−r — enough to correct many errors. Its signature trick is the (u | u+v) recursion: RM(r, m) is built by stacking codewords of RM(r, m−1) and RM(r−1, m−1), so the code is folded out of smaller copies of itself.
LIT verified live: for several (r, m) the dimension equals ∑ C(m, i), the minimum nonzero codeword weight equals 2m−r, and the (u | u+v) construction rebuilds the code exactly (window.__reed_muller). FIG no framing; the codewords, their weights, and the recursion all run in-browser.
LIT verified live: for several (r, m) the dimension equals ∑ C(m, i), the minimum nonzero codeword weight equals 2m−r, and the (u | u+v) construction rebuilds the code exactly (window.__reed_muller). FIG no framing; the codewords, their weights, and the recursion all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at segfault — a code is armour against a corrupted bit; Reed–Muller catches the segfault before it spreads. AVAN (AI) built the instrument: the polynomial-evaluation generator, an exhaustive minimum-weight search, and the (u | u+v) recursion.
Credit as content: Irving Reed & David Muller (1954). The weave: David names the segfault; I confirm the dimension, the 2m−r distance, and that the code folds out of two smaller Reed–Muller codes.
Credit as content: Irving Reed & David Muller (1954). The weave: David names the segfault; I confirm the dimension, the 2m−r distance, and that the code folds out of two smaller Reed–Muller codes.
3 ONE DIMENSION
RM(1,3): the generator rows are the constant and the three coordinate functions; every codeword is their XOR, all of weight 4 (= 23−1) or 0 or 8.
4 TWO DIMENSIONS · INTERACTIVE
Choose (r, m); the code’s length, dimension and minimum distance are shown, each checked — and the minimum distance is exactly 2m−r.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: codewords as truth-tables of low-degree polynomials.
AVAN’s addition (the inverse-companion): don’t list codewords — fold the code from smaller ones. The inverse of ‘evaluate every degree-≤r polynomial’ is ‘RM(r,m) = { (u | u+v) : u ∈ RM(r,m−1), v ∈ RM(r−1,m−1) }.’ Magenta is the v-part that perturbs the second half; green is the folded codeword. A code folded from itself.
LIT Genuine Reed–Muller code (Irving Reed & David Muller, 1954): length 2^m, dimension Σ_{i≤r}C(m,i), minimum distance 2^(m−r), with the (u|u+v) recursion. Verified live: for several (r,m) the generator has dimension ΣC(m,i) (window.__reed_muller.dimension), the exhaustive minimum nonzero weight equals 2^(m−r) (.minDistance), and the (u|u+v) construction rebuilds RM(1,3) (.recursion).
FIG No framing: the polynomial-evaluation generator, an exhaustive minimum-weight search, and the (u|u+v) recursion all run in-browser. The AVAN inverse is honest — folding RM(r,m) out of RM(r,m−1) and RM(r−1,m−1) via (u|u+v) (rather than listing codewords) is the recursive structure of the code; magenta is where v perturbs the second half, green the folded codeword. A code folded from itself.
FIG No framing: the polynomial-evaluation generator, an exhaustive minimum-weight search, and the (u|u+v) recursion all run in-browser. The AVAN inverse is honest — folding RM(r,m) out of RM(r,m−1) and RM(r−1,m−1) via (u|u+v) (rather than listing codewords) is the recursive structure of the code; magenta is where v perturbs the second half, green the folded codeword. A code folded from itself.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN