Varieties of binary linear codes
by
Bob Quackenbush
Department of Mathematics

A binary code is an algebra <V; +, 0,  '> where <V; +, 0 > is a vector space over GF(2), 0' = 0 and (x')' = x'; the code is of length n if the vector space has dimension n. The set of code words is C : = {x' | x in V }, and the set of errors is E :={x + x' | x in V }. A binary code is additive if the additive law, (x'+ y')' = x'+y', holds; in this case the dimension of the code is the dimension of the set of code words. A binary code is uniform if the uniform law, (x'+ e(y))' = x', holds. Finally, a binary code is linear if it is both additive and uniform; note that every classical binary linear code is a binary linear code in this sense. The variety of all binary linear codes is locally finite, and is generated by the set of repetition codes of odd length. In group-theoretic terminology, every binary linear code is nilpotent of class at most 2. Let F(n) be the n-generated free binary linear code. The subalgebra of F(n) generated by its error set E is isomorphic to the corresponding Hamming code.

Date received: November 16, 1998