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Let T be a skew-symmetric Toeplitz matrix with entries in a finite field. For all positive integers n let T n be the upper nxn corner of T, with nullity ν n =ν(T n ). The sequence {ν n :n N} satisfies a unimodality property and is eventually periodic if the entries of T satisfy a periodicity condition. We compute the maximum value and the period of the nullity sequence...