Witt’s extension theorem for mod four valued quadratic forms

Abstract

The mod 4 \bmod \,4 valued quadratic forms defined by E. H. Brown, Jr. are studied. A classification theorem is proven which states that these forms are determined by two things: whether or not their associated bilinear form is alternating, and the σ \sigma -invariant of Brown (which generalizes the Arf invariant of an ordinary quadratic form). Particular attention is paid to a generalization of Witt’s extension theorem for quadratic forms. Some applications to selforthogonal codes are sketched, and an exposition of some unpublished work of E. Prange on Witt’s theorem is provided in an appendix.