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In this paper, we give a sufficient and necessary condition that a twisted Reed-Solomon (TRS) code is MDS. Then we give a sufficient and necessary condition that a twisted generalized Reed-Solomon (TGRS) code is self-dual. Moreover, we present some new explicit constructions of self-dual TGRS codes. These self-dual TGRS codes are MDS, Near-MDS, or 2-MDS and most of them are non-GRS. [ABSTRACT FROM AUTHOR]

Let p≡1(mod4)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\equiv 1\pmod 4$$\end{document} be a prime. In this paper, we support a new method, i.e., a product of 2-adic values for four binary sequences, to determine the maximum evaluations of the 2-adic complexity in all almost balanced cyclotomic binary sequences of order four with period N=p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N=p$$\end{document}, which are viewed as generalizing the results in Hu (IEEE Trans. Inf. Theory 60:5803–5804, 2014) and Xiong et al. (IEEE Trans. Inf. Theory 60:2399–2406, 2014) without the autocorrelation values of cyclotomic binary sequences of order four with period p. By number theory we obtain two necessary and sufficient conditions about the 2-adic complexity of all balanced cyclotomic binary sequences of order four with period N=2p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N=2p$$\end{document} and show the 2-adic complexity of several non-balanced cyclotomic binary sequences of order four with period 2p, which are viewed as generalizing the results in Zhang et al. (IEEE Trans. Inf. Theory 66:4613–4620, 2020). [ABSTRACT FROM AUTHOR]

The Σ REE in loesses of different ages in the Luochuan section shows a narrow range of variation, indicating the homogeneity of loessic materials in chemical composition. The REEs in loess are concentrated mainly in silt-sized detrital minerals. Loesses and paleosols of different ages all are relatively enriched in rare-earth elements of the Ce family, and show similar REE distribution patterns. The fractionation among various REEs in the loess is different from that in morainic, marine and lacustrine sediments, but is similar to that in sand samples from deserts in Northwest China. The ratios of Ce/Ce* and Eu/Eu* reflect that the provenance of loessic materials and their accumulating area are all in an oxidation environment with weakly basic mediums under arid or semi-arid climate. [ABSTRACT FROM AUTHOR]