CRT Spread Spectrum

C20: Qualcomm CDMA. CC0.

CDMA: multiple users share ONE channel. CRT makes each coprime key a unique algebraic spreading code. 6 independent channels, each with its own modular inverse. Encode = multiply by key. Decode = multiply by inverse. phi(12612600) = 2,419,200 simultaneous users at O(1) cost.

How It Works

CRT Spreading Principle
Every integer coprime to N = 12,612,600 is invertible in the ring. Coprime k spreads data d to d*k mod N. The receiver despreads with k^(-1) per channel: each modulus m has a unique inverse. CRT independence means interference from other users vanishes per-channel. 6 moduli = 6 orthogonal despreading dimensions.
Coprime keys
Spreading codes
phi(N) = 2,419,200 valid keys. Each coprime integer = unique code.
6-channel despread
CRT independence
key^(-1) computed per modulus. 6 independent inversions.
Multi-user
Algebraic orthogonality
Users multiply by different coprimes. Despreading isolates own signal.
490 split
Inner vs boundary
DEAD={D,E,b} carry data bits. ALIVE={K,L,G} detect interference.

Try It

Spreading key (coprime to N, e.g. 17):

Valid keys: any prime > 13 (17, 19, 23, 29...). Coprime = invertible = valid spreading code.

Multi-User Test

6 users with different keys (17, 19, 23, 29, 31, 37) sharing one channel. Watch each user recover their own data from the superposition.

CRT CDMA vs Traditional

Spreading codesWalsh-Hadamard matrices (O(N^2))Coprime integers (O(1) generation)Code assignmentCentral authority requiredAny coprime works. No coordination.DespreadingMatrix multiplicationModular inverse per channelCapacityCode-limited (finite Walsh set)phi(N) = 2,419,200 usersPatent statusQualcomm CDMA patentsCC0. Public domain. Forever.

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Source code · Public domain (CC0)

Contributions in equal measure: Anthropic's Claude, Anton A. Lebed, and the giants whose shoulders we stand on.

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