The research currently present on this page relates to the work I did at Trinity University in San Antonio, Texas during the summer of 2001. Our work focused on the spectrum of cross numbers of minimal zero sequences in finite Abelian groups, primarily in Z_2p^n and Z_pq. A quick overview of the basic concepts in this branch of additive number theory is given at the beginning of our paper, which is found below.
Paul Baginski, Scott T. Chapman, Kathryn McDonald, and Lara Pudwell. On Cross Numbers of Minimal Zero Sequences In Certain Cyclic Groups Awaiting publication in Ars Combinatoria
We have gathered extensive data on the spectra of Z_pq for p and q distinct primes, and have made the majority of it available below. Some files were too large (5MB+) to be placed on the webpage and are available upon request.
Data Set 1
The first set of data corresponds to Lemma 4.1 in our paper. The program simply generates the subsets predicted by this lemma and returns the contiguous gaps, along with the size of each gap. Note that some of these "missing" values given by the program may actually be cross numbers of minimal zero sequences; the lemma is solely a lower bound on W(Z_pq). Surprisingly, for all cases we have considered, the values generated by the lemma have been all the possible values, leading us to conjecture that perhaps the family of minimal zero sequences created prior to Lemma 4.1 form a complete family with regard to generating W(Z_pq). A link to the program, source code, and basic instructions on how to run the program are given here. Pregenerated data is organized below.
The other set of data calculates the minimal zero sequences for a given p and q (with p greater than q) in a brute force manner. This program does determine W(Z_pq) completely. However, due to the exponential nature of the cardinality of the set of minimal zero sequences, this program is extremely inefficient in practice. We have made use of a few facts to facilitate the speed of the program, namely:
The structure of W(Z_pq) is determined for values less than one, so we need only generate minimal zero sequences with cross number greater than 1.
Cross numbers of minimal zero sequences are invariant under automorphisms of the group.
Every minimal zero sequence in Z_pq with cross number greater than one must contain an element whose order is the order of the group.
These facts show that it suffices to generate the family of minimal zero sequences that contain an occurence of the element 1. This family generates all the possible cross numbers greater than 1, and any minimal zero sequence with cross number greater than 1 is the image of one of these sequences under an automorphism. Note that the family generated usually contains several members that are automorphic images of each other, so these minimal zero sequences do not form a complete set of representatives of the equivalence classes of minimal zero sequences related by automorphism.
The program, its source code, and basic instructions on how to run the program can be found here. Pregenerated data is organized below.
(c) All material accessible through this page is copyright by Paul Baginski and his coauthors. Permission is granted for fair use in personal, noncommercial, and academic projects.
Paul Baginski, Scott T. Chapman, Kathryn McDonald, and Lara Pudwell. On Cross Numbers of Minimal Zero Sequences In Certain Cyclic Groups Awaiting publication in Ars Combinatoria