The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, 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, and minimum distance. We also obtain their dual codes. 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 et al. (IEEE Trans Inf Theory 64(2):900–908, 2018) when applied to the one-point AG codes over the extended norm-trace curve.

递减范数迹代码是由一组可整除的单项式和扩展的范数迹曲线的有理点定义的评估代码。特别是，递减的范数迹代码包含扩展范数迹曲线上的单点代数几何（AG）代码。我们利用Gröbner基理论并找到曲线有理点上的指示函数来确定递减范数迹码的基本参数：长度、尺寸和最小距离。我们还获得了他们的双重代码。我们给出了递减范数迹码成为自正交码或自对偶码的条件。我们提供了一种线性精确修复方案来纠正单次擦除以减少范数跟踪代码，该方案适用于比 Jin 等人开发的方案更高速率的代码。（IEEE Trans Inf Theory 64(2):900–908, 2018）应用于扩展范数轨迹曲线上的单点 AG 代码时。