Binary construction of quantum codes of minimum distance three and four

We give elementary recursive constructions of binary self-orthogonal codes with dual distance four for all even lengths n≥12 and n=8. Consequently, good quantum codes of minimum distance three and four for such length n are obtained via Steane´s construction and the CSS construction. Previously, such quantum codes were explicitly constructed only for a sparse set of lengths. Almost all of our quantum codes of minimum distance three are optimal or near optimal, and some of our minimum-distance four quantum codes are better than or comparable with those known before.