Codes and Polynomials in the Study of Cyclic Difference Sets

Citation

Norwood, Thomas E. (1996) Codes and Polynomials in the Study of Cyclic Difference Sets. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/sjay-8s96. https://resolver.caltech.edu/CaltechTHESIS:07142025-170333525

Abstract

Let II_q be the Desarguesian projective plane of order q = 2^m. We define an incidence structure as follows. Let O be a regular oval in II_q and let P be the set of exterior points of O. For each p Ɛ P define B_p to be the sum (mod 2) of the exterior blocks through p. Then the B_p are the blocks of a (q² - 1, q²/2,q²/4) cyclic difference set which we denote M(q²). It was conjectured by Assmus and Key in [ass3] and [ass1] that rank₂(C₂(M(q²))) = m2^m-1. The goal of this paper is to give a proof of that conjecture as well as to discuss certain related results which are suggested by it or by its proof.

There are three central theorems in this paper. The first is theorem 2.2.3: rank₂(C₂(M(q²))) = m2^m-1, resolving the conjecture of Assmus and Key. Although the proof of this theorem does not directly involve the cyclic nature of M(q²), we do utilize some results and a construction of Jackson [jac] on designs with PSL₂(q) acting transitively. Thus, it is of interest to us to undertake a further study of Jackson's construction and its relation to cyclic difference sets. This gives rise to our second major result, that Jackson's construction is equivalent to a classical construction of Gordon, Mills, and Welch [gor]. This is the primary result of chapter 1 and an immediate consequence is theorem 1.3.5, that PSL₂(q) acts transitively on a Hadamard design D if and only if D arises from the Gordon, Mills, Welch construction. Another particularly interesting consequence of the equivalence of the two constructions is that although the designs M(q²) and a certain family of the Gordon, Mills, Welch designs are isomorphic, this has apparently not been noticed until now even though both families have been widely studied. The third major theorem of this paper is theorem 3.1.1 which characterizes the generator polynomial of the binary code of M(q²) by explicitly giving its roots. The proof of this theorem comprises the bulk of chapter 3. The most important application of this theorem is that it allows us to study the code C₂(M(q²))) as a cyclic code. That is, we may immediately determine, from the roots, exactly which cyclic codes are subcodes of C₂(M(q²))). In particular, we address the question of whether it contains a cyclic punctured first-order Reed Muller code in theorem 3.2.1. This question was also posed by Assmus and Key in [ass1], and [ass3].

Finally, in chapter 4, we discuss a generalization of the results of the earlier chapters. Specifically, let C_rep(v) be the subcode of F^v₂ generated by repetition vectors (i.e. vectors of the form (c, c,...,c)). If v = 2^m - 1, a cyclic code is a subcode of C_rep(v) if and only if it does not contain a cyclic punctured first-order Reed Muller code. For this reason, we propose that the question of containment in C_rep(v) is of interest for the code of any cyclic difference set, not just those with v = 2^m - 1. In chapter 4, we address this question in the case of all known cyclic Hadamard difference sets, as well as the Singer difference sets. In the latter case, the codes are given by generalized Reed Muller codes, so we determine the relation of these codes to C_rep(v).

Thesis Files