1. A method for controlling errors in an electronically recorded digital data message by performing at least one of a plurality of predetermined arithmetic operations on the data message in one or more of a plurality of subfields of a first GF(2.sup.(m)(p)) or a second GF(2.sup.(m)(q)) finite field, using an arithmetic unit including a plurality of multiplier gates and a plurality of addition gates, said method comprising steps of: (a) factoring a first, second, and third composite number (m)(p), (m)(q), (p)(q) respectively wherein the first and second composite numbers are the number of bits designating each element in the first GF(2.sup.(m)(p)) and second GF(2.sup.(m)(q)) finite fields; (b) forming a first primitive polynomial cubically extending each element in a first subfield GF(2.sup.(m)) to the first finite field GF(2.sup.(m)(p)), forming a second primitive polynomial quadratically extending each element in the first subfield GF(2.sup.(m)) to the second finite field GF(2.sup.(m)(q)), q.noteq.p, and forming a third primitive polynomial for quadratically extending each element in a second subfield GF(2.sup.(p)(q)) to the first finite field GF(2.sup.(m(p)); and (c) performing at least one of the plurality of predetermined arithmetic operations on the data message by utilizing the arithmetic unit coupled to receive the data message, said arithmetic operations being selected either from a first group of operations associated with the first subfield as cubically extended to the first finite field or as quadratically extended to the second finite field, or selected from a second group of operations associated with the second subfield as quadratically extended to the first finite field, wherein the selected arithmetic operation is performed in the associated subfield.