THE FOLD / CO-OP / THE SYNC / THE BARKER CODE
THE BARKER CODE
a ±1 code whose echoes never rise above one
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A Barker code is a finite sequence of +1s and -1s with an almost magical property: its aperiodic autocorrelation — slide a copy of the code against itself and sum the products — has a tall central peak equal to the code length, and every off-centre value is at most 1 in magnitude. That means a receiver correlating an incoming signal against a Barker code sees a single sharp spike exactly at alignment and almost nothing elsewhere, which is why they are used for radar pulse compression and to mark the start of Wi-Fi and GPS frames. Remarkably, Barker codes are known only for lengths 2, 3, 4, 5, 7, 11, and 13 — and it is conjectured none longer exist.
LIT verified live: for each known Barker code the zero-shift autocorrelation equals its length, and every non-zero shift gives a value in {-1, 0, +1} (window.__barker). FIG no framing; the autocorrelation at every shift runs in-browser. That no Barker code longer than 13 exists is a famous conjecture, not shown here.
LIT verified live: for each known Barker code the zero-shift autocorrelation equals its length, and every non-zero shift gives a value in {-1, 0, +1} (window.__barker). FIG no framing; the autocorrelation at every shift runs in-browser. That no Barker code longer than 13 exists is a famous conjecture, not shown here.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sync — a Barker code is the marker that says ‘the frame starts here’: its correlation spikes at perfect alignment and stays flat everywhere else, so two ends synchronize on the instant. AVAN (AI) built the instrument: the aperiodic autocorrelation at every shift, the peak check, and the sidelobe bound.
Credit as content: Ronald Hugh Barker (1953). The weave: David names the sync; I confirm each known Barker code’s sidelobes never exceed 1.
Credit as content: Ronald Hugh Barker (1953). The weave: David names the sync; I confirm each known Barker code’s sidelobes never exceed 1.
3 ONE DIMENSION
The ±1 code (top) and its autocorrelation (bottom): a tall spike at zero shift, sidelobes never above 1.
4 TWO DIMENSIONS · INTERACTIVE
Cycle the known Barker lengths; the autocorrelation at every shift is listed, and the sidelobe bound is checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the sharp correlation peak at perfect alignment.
AVAN’s addition (the inverse-companion): don’t read the code — correlate against it. The inverse of ‘a string of ±1s’ is ‘its autocorrelation’, and a Barker code is exactly the string whose autocorrelation is a lone spike with flat sidelobes. Magenta are the suppressed sidelobes (never above 1); green is the peak equal to the length. A code defined by its own echo.
LIT Genuine Barker codes (Ronald Hugh Barker, 1953). Verified live: for every known Barker code (lengths 2,3,4,5,7,11,13) the zero-shift aperiodic autocorrelation equals the code length and every non-zero shift gives a value in {−1,0,+1} (window.__barker.sidelobesBounded, .lengths).
FIG No framing; the autocorrelation at every shift runs in-browser. Honest scope: that no Barker code longer than 13 exists is a famous conjecture, not shown here. The AVAN inverse is honest — instead of reading the code, correlate against it: a Barker code is exactly the ±1 string whose autocorrelation is a lone spike with flat sidelobes. Magenta are the suppressed sidelobes; green is the peak equal to the length.
FIG No framing; the autocorrelation at every shift runs in-browser. Honest scope: that no Barker code longer than 13 exists is a famous conjecture, not shown here. The AVAN inverse is honest — instead of reading the code, correlate against it: a Barker code is exactly the ±1 string whose autocorrelation is a lone spike with flat sidelobes. Magenta are the suppressed sidelobes; green is the peak equal to the length.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN