María Concepción López-Díaz
Universidad de Oviedo, Spain
Galois ring valued quadratic forms.
Abstract: Galois Rings were first studied by W. Krull more than 80 years ago and rediscovered independently by G.J. Janusz and R. Raghavendran in the decade of the 60's. However, these rings have received a major attention in the last years due to their nice applications to Coding Theory and Cryptography. For instance, the $\mathbb{Z}_4$-valued quadratic forms defined by E. H. Brown and studied by J. A. Wood have been used to produce different families of non equivalent binary Kerdock codes. An invariant was included by E.H. Brown along with the definition of $\mathbb{Z}_4$-valued quadratic forms. J.A. Wood proved later that this invariant classifies, together with the type of the corresponding bilinear form (alternating or not), nonsingular $\mathbb{Z}_4$-valued quadratic forms. We consider quadratic forms valued in a Galois Ring of arbitrary characteristic and show that their study can be reduced to the case of characteristic $p^2$. In this case we introduce and study the main properties of an invariant, that is considered in a Galois Ring of characteristic $p^3$. This invariant classifies, together with the type of the corresponding bilinear form (alternating or not), nonsingular Galois Ring valued quadratic form. Moreover, simple proofs of Witt's Cancelation and Extension Theorems follow from the study of this invariant. This remedy the limitations in the exposition of Wood's results and can be applied to the quadratic forms taking values in a Galois Ring of characteristic 4 used to construct different families of non necessarily equivalent Generalized Kerdock codes over a finite field of characteristic 2.