Decreasing norm-trace codes
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This paper introduces decreasing norm-trace codes, which are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. As a particular case, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve. We use Gröbner basis theory and find the indicator functions on the rational points of the curve to determine the basic parameters of the decreasing norm-trace codes: length, dimension, minimum distance, and the dual code. We give conditions for a decreasing norm-trace code to be a self-orthogonal or a self-dual code. We provide a linear exact repair scheme to correct single erasures for decreasing norm-trace codes, which applies to higher rate codes than the scheme developed by Jin, Luo, and Xing (IEEE Transactions in Information Theory 64 (2), 900-908, 2018) for the one-point AG codes over the extended norm-trace curve.
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