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Katz, Philippe Langevin, Sangman Lee, Yakov Sapozhnikov","submitted_at":"2016-08-14T01:52:26Z","abstract_excerpt":"We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=\\sum_{x \\in F} \\psi(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $\\psi$ is the canonical additive character of $F$, the exponent $d$ is relatively prime to $|F^\\times|$, and $a$ is an element of $F$. Such sums often arise in arithmetical calculations and also have applications in information theory. For each $F$ and $d$ one would like to know $V_{F,d}$, the minimum $p$-adic valuation of $W_{F,d}(a)$ as $a$ runs through the elements of $F$. 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