1. A data processing system for determining at least one of locations and magnitudes of errors in a received code word via respective coefficients of at least part of one of an error-locator polynomial and an error-evaluator polynomial of degree t in at least part of an inversion-free Berlekamp-Massey algorithm, wherein t is an integer, the system comprising: a plurality of arithmetic units, each arithmetic unit comprising a finite field multiplier circuit and at least one finite field adder circuit, the finite field multiplier circuit having an output coupled to an input of said at least one finite field adder circuit for selectively performing at least two finite field arithmetic calculations between values associated with said received code word presented at two inputs to the finite field multiplier circuit and a further value associated with said received code word presented at two inputs to the finite field adder circuit, one of said inputs to the finite field adder circuit being coupled to the output of the finite field multiplier circuit and the other of said inputs to the finite field adder circuit being adapted to receive further input data, such that the data processing system includes no more than (t+1) finite field multiplier circuits; memory for storing a previous finite field arithmetic calculation result of a first arithmetic unit of the plurality of arithmetic units for use in a current finite field arithmetic calculation of the first arithmetic unit; and at least one finite field adder circuit for combining respective finite field arithmetic calculation results of respective current finite field arithmetic calculations of at least two of the arithmetic units.