Various constructions of good codes (Q2704373)

This is a survey article on applications of number theory and algebraic geometry to coding theory. First, the results from the 1980s on good algebraic-geometry codes are reviewed. Then the method of class field towers for the construction of asymptotically good towers of global function fields is described, which leads in particular to the result of \textit{H. Niederreiter} and \textit{C. P. Xing} [Math. Nachr. 195, 171-186 (1998; Zbl 0920.11039)] on sequences of algebraic-geometry codes beating the Gilbert-Varshamov bound. Further topics include codes over Galois rings, e.g. the well-known construction of good binary nonlinear codes from linear codes over \({\mathbb Z}_4\), and character sum techniques for measuring the distribution of symbols in nonzero codewords of certain codes.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00031].