Spurious Galois fields (coding theory applications)

The algebraic system consisting of a finite set of elements with two internal operations of addition and multiplication is studied. For the above system, which satisfies all the axioms of the fields except for the axiom of associativity, which may, but need not, be satisfied, the name spurious Galois field and the symbol SGF(q) has been proposed. Such an approach makes it possible to show that the spurious Galois fields form a class of algebraic systems containing all the isomorphic Galois fields as its smaller subclass. Some properties of SGF(q) are presented, and some domains of their applications are pointed out.