Construction of self-dual normal bases and their complexity

Recent work of Pickett has given a construction of self-dual normal bases for extensions of finite fields, whenever they exist. In this article we present these results in an explicit and constructive manner and apply them, through computer search, to identify the lowest complexity of self-dual normal bases for extensions of low degree. Comparisons to similar searches amongst normal bases show that the lowest complexity is often achieved from a self-dual normal basis

Éléments explicites en théorie algébrique des nombres

This report consists in a synthesis of my research activities in algebraic number theory, between 2003 and 2013, on my own or with colleagues. The main goal is the study of the Galois module structure of modules associated to number field extensions, under various hypothesis, specifically about ramification. We also present results about other subjects which came into the way of the previous study: the construction of a certain type of Galois extensions of the field of rationals, the complexity of self-dual normal bases for multiplication in finite fields, and a bit of combinatorics. We stress the importance of an explicit knowledge of the objects under study.Ce mémoire présente une synthèse de mes travaux de recherche en théorie algébrique des nombres menés entre 2003 et 2013, seul ou en collaboration. Ils portent principalement sur l'étude de la structure galoisienne de modules associés à des extensions de corps de nombres, sous diverses hypothèses en particulier de ramification. Ils abordent aussi des thèmes rencontrés chemin faisant : construction d'un certain type d'extensions galoisiennes du corps des rationnels, complexité des bases normales auto-duales pour la multiplication dans les corps finis, un peu de combinatoire. Dans la présentation de tous ces travaux, l'accent est mis sur l'aspect explicite des objets étudiés

Counting and constructing orthogonal circulants

AbstractIf F is an arbitrary finite field and T is an n × n orthogonal matrix with entries in F then one may ask how to find all the orthogonal matrices belonging to the algebra F[T] and one may want to know the cardinality of this group. We present here a means of constructing this group of orthogonal matrices given the complete factorization of the minimal polynomial of T over F. As a corollary of this construction scheme we give an explicit formula for the number of n × n orthogonal circulant matrices over GF(pl) and a similar formula for symmetric circulants. These generalize results of MacWilliams, J. Combinatorial Theory10 (1971), 1–17