BCH CODES and BOUNDS FOR CYCLIC CODES

Abstract

A monic polynomial g(x)∈ GF(q)[x] is said to split in the extension field GF(q m ) of GF(q) if g(x) can be factored as a product of linear polynomials in GF(q m ), i.e. if we can write $$ g(x) = (x - {\alpha _1})(x - {\alpha _2})...(x - {\alpha _n}) $$ where αi ∈ GF(q m ). Here GF(q m ) is called a splitting field of g(x).In general, we refer to the splitting field of a polynomial g(x)∈ GF(q)[x] as the smallest field GF(q m )in which g(x) splits, or in other words, the smallest field containing all roots of g(x). It is not difficult to prove that the splitting field of a polynomial always exists (e.g. see [Lidl 84]). Indeed, the splitting field of g(x) can be determined from the degrees of its irreducible factors over GF(q) (see exer­cise 29). Note that g(x) may be irreducible over GF(q), but (always) factors as a product of dis­tinct linear polynomials in its splitting field. For example, g(x) = x 2 + x + 1 is irreducible over GF(2) and has no roots in GF(2); but over GF(4), $$ g(x) = (x + \alpha )(x + {\alpha ^2}) $$ and has roots α and α2, where GF(4) = {0,l,α,α2} with α2 + α + 1 = 0.