2934 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 7, NOVEMBER 2001 Almost Difference Sets and Their Sequences With Optimal Autocorrelation K. T. Arasu, Cunsheng Ding, Member, IEEE, Tor Helleseth, Fellow, IEEE, P. Vijay Kumar, Senior Member, IEEE, and Halvard M. Martinsen Abstract—Almost difference sets have interesting applications in cryptography and coding theory. In this paper, we give a well- rounded treatment of known families of almost difference sets, es- tablish relations between some difference sets and some almost dif- ference sets, and determine the numerical multiplier group of some families of almost difference sets. We also construct six new classes of almost difference sets, and four classes of binary sequences of period with optimal autocorrelation. We have also obtained two classes of relative difference sets and four classes of divisible difference sets (DDSs). We also point out that a result due to Jungnickel can be used to construct almost difference sets and sequences of period with optimal autocorrelation. Index Terms—Almost difference sets, correlation, cyclotomy, difference sets, divisible difference sets (DDSs), relative difference sets, sequence. I. INTRODUCTION L ET be an Abelian group of order . Let be a -subset of . The set is an almost differ- ence set of if takes on the value altogether times and the value altogether times when ranges over all the nonzero elements of , where is the differ- ence function defined by and Let be a group of order and a subgroup of of order . If is a -subset of , then is called an divisible difference set (DDS) provided that Manuscript received July 6, 2000; revised May 30, 2001. The work of K. T. Arasu was supported in part by the National Science Foundation under Grant CCR-9814106 and by NSA under Grant 904-01-1-0041. K. T. Arasu is with the Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435 USA (e-mail: karasu@math.wright.edu). C. Ding is with the Department of Computer Science, Hong Kong Univer- sity of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (e-mail: cding@cs.ust.hk). T. Helleseth and H. M. Martinsen are with the Department of Informatics, University of Bergen, HIB, N-5020 Bergen, Norway (e-mail: torh@ii.uib.no; halvard@ii.uib.no). P. V. Kumar is with Communication Science Institute, Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089-2565 USA (e-mail: vijay@ceng.usc.edu). Communicated by A.M. Klapper, Associate Editor for Sequences. Publisher Item Identifier S 0018-9448(01)08584-4. the list of differences contain every nonidentity element of exactly times and every ele- ment of exactly times. If , is called a relative difference set, and is called the forbidden subgroup. Davis [3] called a DDS an almost difference set if and differ by . Hence, the almost difference sets defined by Davis are a special class of the almost difference sets above. Davis defined this special class of almost difference sets due to its relationship to symmetric difference sets [3]. Another kind of almost difference sets were defined by Ding [6]–[8] (see also [2, p. 140]) for the study of cryptographic func- tions with optimal nonlinearity. Ding, Helleseth, and Lam have considered this special class of almost difference sets for con- structing binary sequences with three-level autocorrelation [9]. In fact, the special class of almost difference sets defined by Ding are actually almost difference sets, which are only defined for odd . Ding, Helleseth, and Martinsen [10] have generalized the two kinds of almost difference sets by defining the al- most difference sets, for the purpose of obtaining binary se- quences with optimal autocorrelation. This broader class of al- most difference sets was studied independently by Mertens and Bessenrodt for the Bernasconi model in physics [21]. It is nice that the current almost difference sets unify the two different kinds of almost difference sets introduced by Davis and Ding, respectively. Clearly, the special class of almost dif- ference sets introduced by Davis are a subclass of DDSs, while there are almost difference sets that are not DDSs. Almost difference sets are closely related to cryptography [2], coding theory, and sequences [9], [10]. They can be used to construct cryptographic functions with optimal nonlinearity, se- quences with optimal autocorrelation, and good constant-weight codes. So far only a small number of classes of al- most difference sets have been discovered. In this paper, we give a well-rounded treatment of known fam- ilies of almost difference sets, establish relations between some difference sets and some almost difference sets, and determine the numerical multiplier group of some families of almost difference sets. We also construct six new classes of almost difference sets, and four classes of binary sequences of period with optimal autocorrelation. We have also obtained two classes of relative difference sets and four classes of DDSs. We also point out that a result due to Jung- nickel can be used to construct almost difference sets and se- quences of period with optimal autocorrelation. 0018–9448/01$10.00 © 2001 IEEE