where coefficients ( \alpha_{i,j} ) form a ( \alpha_{i,j} = \xi_i^{,j} ) with distinct ( \xi_i \in GF(2^q) ) or real numbers.
: Choose primitive element ( \xi \in GF(8) ), set ( \xi_i = \xi^i ). Track ( i ) sends ( x_i[t] = Tr(\xi_i^0) \cdot 1 + Tr(\xi_i^1) \cdot t + Tr(\xi_i^2) \cdot t^2 ) (over GF(2), addition is XOR). This yields 4 distinct binary sequences of period ( 2^3-1=7 ). 4. Synchronization and Skew Estimation At the receiver, for each track ( i ) we collect ( L ) samples. Cross-correlate track ( i ) with track ( 0 ) (reference) using the known polynomial structure. poly track track codes
[ x_i[t] = \sum_{j=0}^{m-1} \alpha_{i,j} \cdot b_j[t] ] where coefficients ( \alpha_{i,j} ) form a (
(Skew estimation bound): With ( L \geq 2 \cdot \max(\tau_i) + \text{poly}(m) ), the probability of skew estimation error ( < 2^{-L} ) for AWGN channels with SNR > 6 dB. 6. Application Example: Polygraph Chart Alignment In analog polygraph recording (breathing, GSR, heart rate, blood pressure), each pen writes on a moving paper chart. Over time, mechanical drift causes trace skew — the same time event appears at different horizontal positions on different traces. This yields 4 distinct binary sequences of period