Regarding convolutional codes as polynomial analogues of arithmetic lattices, we derive a Poisson Jacobi formula for their trivariate weight enumerator. The proof is based on harmonic analysis on locally compact abelian groups as developed in Tate's thesis to derive the functional equation of the zeta function.

Families of binary low-correlation sequences with high nonlinearity (in relation with their Walsh Hadamard transform) are constructed by using the most significant bit of linear recurrence sequences over the ring Zopf2l, for lges3. The engineering motivation is the design of a multiple-code code-division multiple-access (CDMA) scheme with a control of low peak-to-average power ratio (PAPR). Proof techniques combine Galois ring theory (local Weil bound) with spectral analysis over the additive group of Zopf2l. New estimates on the size of weighted degree trace codes are derived. The parameters of the sequences families constructed are shown to lie above a modified Varshamov-Gilbert bound

The imbalance and the autocorrelation of the binary sequences in the title are explored by combining the local Weil bound with spectral analysis. Recent estimates on these quantities are improved by a factor of order 2/sup l/2/, for large l.

Using known techniques of desingularization for singular plane projective curves over finite fields F/sub q/, q=2/sup m/, we found several new binary plane projective curves of genus 3, 4, and 5 with the maximal number of F/sub q/-rational points (q=2/sup m/, m=3, 4, 5, 6, 7, 8, and 9) on their smooth projective models, which are close to or meet Serre's upper bound.