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In this paper, by calculating the dual code of the Schur square for the standard twisted Reed-Solomon code, we give a sufficient and necessary condition for the generalized twisted Reed-Solomon code with h + t ≤ k − 1 to be self-orthogonal, where k is dimension, h is hook and t is twist. And then, we show that there is no self-orthogonal generalized twisted Reed-Solomon code under some conditions. Furthermore, several classes of self-orthogonal generalized twisted Reed-Solomon codes are constructed, and some of these codes are non-GRS self-orthogonal MDS codes or NMDS codes.

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In this paper, by using some properties for linear algebra methods, the paritycheck matrixs for twisted generalized Reed-Solomon codes with any given hook h and twist t are presented, and then a sufficient and necessary condition for that a twisted generalized Reed-Solomon code with dimension h + t (h ≥ t) to be self-dual is given. Furthermore, several classes of self-dual codes with small singleton defect are constructed based on twisted generalized Reed-Solomon codes.