Linear block codes for four-dimensional signals
The author studies linear codes over the field \(Z[\xi_n]/ \langle a_0+a_1 \xi_n+a_2 \xi^2_n+ a_3\xi^3_n \rangle\) where \(a_0+a_1 \xi_n+a_2 \xi^2_n+ a_3\xi^3_n\) is an irreducible element of \(Z[\xi_n]\), and \(\xi_n\) is a \(n\)-th root of unity for \(n=5,8\) and 12. The order of the field is connected to the norm of \(a_0+a_1 \xi_n+ a_2\xi^2_n+ a_3\xi^3_n\). The elements of the field \(Z[\xi_n]/\langle a_0+ a_1\xi_n+ a_2\xi^2_n+ \xi^3_n\rangle\) can be written as 4-tuples (i.e. coefficients of \(1,\xi_n,\xi^2_n, \xi^3_n)\), which together form the four-dimensional signal space. The linear codes constructed by the author can correct errors of a specific type. The paper contains some misprints: On page 63 the integers \(p\) of the form \(5k\to 4\), \(5k\to 3\) and \(5k\to 2\) must be \(5k+4\), \(5k+3\) and \(5k+2\) respectively; On page 69 \(R^5_{p^4}\) in the third and fifth line must be replaced by \(R^n_{p^4}\) in Corollary 2.1: the first \(GF(p^2)\) has to be replaced by \(GF(p)\), and in example 3 on page 73 the ring \(Z[\xi_2]\) must be \(Z[\xi_5]\). Furthermore in the example on page 72 the author shifts back and forth between the different representations of the field \(Z[\xi_8]/ \langle 1+\xi_8+ a_2\xi^3_8 \rangle\) which is isomorphic to \(GF(41)\). It is unclear to the reviewer how the syndrom is calculated by the receiver. In order to be able to calculate the syndrom he has to find the appropriate representative for the element \([1,1,1,-1]\) which is not in the table. The author should explain how the right respresentative is obtained.
scientific article; zbMATH DE number 1271777
scientific article