Asymptotically good ZprZpsadditive cyclic codes Abstract   We construct a class of      Z      p    r        Z      p    s       additive cyclic codes generated by pairs of polynomials, where p is a prime number. Based on probabilistic arguments, we determine the asymptotic rates and relative distances of this class of codes: the asymptotic GilbertVarshamov bound at      1      p    s  xe2x88x92  r      2    xcexb4    is greater than      1    2      and the relative distance of the code is convergent to xcexb4, while the rate is convergent to      1    1      p    s  xe2x88x92  r        for    0    xcexb4      1    1      p    s  xe2x88x92  r        and    1  xe2x89xa4  r    s   . As a consequence, we prove that there exist numerous asymptotically good      Z      p    r        Z      p    s       additive cyclic codes.

Highly nonlinear functions over finite fields Abstract   We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from      F    q    n      to      F    q      to the set of affine functions from      F    q    n      to      F    q     . We prove the conjecture for each q such that the characteristic of      F    q      lies in a subset of the primes with density 1 and we prove the conjecture for all q by assuming the generalised Riemann hypothesis. Roughly speaking, we show the existence of functions for which the distance to the affine functions is maximised when n tends to infinity. This also determines the asymptotic behaviour of the covering radius of the        q    n    ,  n    1      ReedMuller code over      F    q      and so answers a question raised by Leducq in 2013. Our results extend the case    q    2   , which was recently proved by the author and which corresponds to the original conjecture by Patterson and Wiedemann. Our proof combines evaluations of Gauss sums in the semiprimitive case, probabilistic arguments, and methods from discrepancy theory.