New Partial Difference Sets in Ztp2 and a Related Problem about Galois Rings
✍ Scribed by Xiang-Dong Hou; Ka Hin Leung; Qing Xiang
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 212 KB
- Volume
- 7
- Category
- Article
- ISSN
- 1071-5797
No coin nor oath required. For personal study only.
✦ Synopsis
We generalize a construction of partial di!erence sets (PDS) by Chen, Ray-Chaudhuri, and Xiang through a study of the TeichmuK ller sets of the Galois rings. Let R"GR(p, t) be the Galois ring of characteristic p and rank t with TeichmuK ller set ¹ and let : RPR/pR be the natural homomorphism. We give a construction of PDS in R with the parameters v"pR, k"r(pR!1), "pR#r!3r, "r!r, where r"lpR\Q NR , 14l4pQ NR , and s(p, t) is the largest dimension of a GF(p)-subspace =LR/pR such that (=)5¹ generates a subgroup of R of rank(t. We prove that s(p, t) is the largest dimension of a GF(p)-subspace = of GF(pR) such that dim =N(t, where =N is the GF(p)-space generated by +N G w G " w G 3=, 14i4p,. We determine the values of s(p, t) completely and solve a general problem about dim # =P for an E-vector space = in a "nite extension of a "nite "eld E. The PDS constructed here contain the family constructed by Chen, Ray-Chaudhuri, and Xiang and have a wider range of parameters.