IS THE (133,33,8)-DIFFERENCE SET REALLY SPORADIC?

Difference sets are important objects in the theory of combinatorial designs, which can be traced back to Singer's paper published in 1938. Systematic research began in the late 1940s with Hall's analysis of cyclic difference sets.

In Hall's 1956 paper, all cyclic difference sets with parameters (v, k, \lambda) with kin the range 3≤k≤50 were examined by using multipliers. Hall categorized the acquired cyclic difference sets, and identified merely two instances of them that fall out-side the parameters encompassed by the categories he outlined, one of them is the cyclic (133, 33, 8)-difference set, which is the subject of this thesis.

The thesis attempts to generalize the cyclic (133, 33, 8)-difference set using three methods (mainly in two directions, from the points of view of cyclotomic classes and sub-difference sets), but does not find any further nontrivial examples.

Since sub-difference sets of Singer difference sets and two-weight irreducible cyclic codes are equivalent objects, as Schmidt and White gave a conjecturally complete clas-sification of two-weight irreducible cyclic codes, one can obtain the conjectural classifi-cation of all nontrivial sub-difference sets of Singer difference sets, and one of which is the cyclic (133, 33, 8)-difference set. Given the numerical conditions, Schmidt and White verified the cases for u≤100, 000, where u is the index. The thesis improves the upper bound of the conjecture for u≤10, 000, 000, and rules out some small cases of difference sets with Singer parameters.

[1] SINGER J. A THEOREM IN FINITE PROTECTIVE GEOMETRY AND SOME APPLICA-TIONS TO NUMBER THEORY[Z].
[2] HALL M. A Survey of Difference Sets[J]. Proceedings of the American Mathematical Society, 1956, 7(6): 975-986.
[3] SCHMIDT B, WHITE C. All Two-Weight Ireducible Cyclic Codes?[J]. Finite Ficlds and Their Applications, 2002, 8(1): 1-17.
[4] DINITZ J H, STINSON D R. Contemporary design theory : a cllction of surveys[C/OL]// 1992. htpt:/pisemnanticscholar .org/CorpusID:118355962.
[5] STORER T. Cyclotomies and Difference Sets modulo a Product of Two Distinct Odd Primes. [J]. Michigan Mathematical Jourmal, 1967, 14(1).
[6] BRUCK R H, RYSER H J. The Nonexistence of Certain Finite Projctive Planes[J/OL]. Canadian Journal of Mathematics, 1949, 1: 88 - 93. htps:/api. semanticscholar.org/CorpusID: 123440808.
[7] CHOWLA s, RYSER H J. Combinatorial Problems[J/OL]. Canadian Journal of Mathematics, 1950, 2: 93.99. htpt:/api.semanticscholar.or/CorpuslD:247194753.3
[8] RYSER H. The Existence of Symmetric Block Designs[/OL]. J. Comb. Theory, Ser. A, 1982, 32: 103-105. htp:/api. semanticscholar .org/CorpusID:3281 1940.
[9] TURYN R J. Character sums and dfrence sets.[J/OL]. Pacific Jourmal of Mathematics, 1965, 15: 319-346. htps:/apisemanticscholar.org/CorpusID:59478374.
[10] YAMAMOTO K. Decomposition fields of dfference sets[J/OL]. Pacific Journal of Mathematics, 1963, 13: 337-352. ht:/apisemanticscholar.org/CorpusID:120980698.
[11] HALL J. Cyclic projective planes[J/OL]. Duke Mathematical Journal, 1947, 14: 1079-1090. https://apisemanticscholar. org/CorpusID: I 19846649.
[12] BRUCK R H. DIFFERENCE SETS IN A FINITE GROUP[J/OL]. Transactions of the American Mathenatical Society, 1955, 78: 464-481. htpsp//api. semanticscholar .org/CorpusID: 120486510.
[13] MCFARLAND R B, MANN H B. On Multipliers of Difference Sets[J/OL]. Canadian Journal of Mathematics, 1969, 17: 541- 542. htpt:/pi semanticscholar.org/CorpusID:116891713.
[14] MCFARLAND R L, RICE B F Translates and multipliers of abelian diference sets[C/OL]// 1978. htps:/pisemanticscholar .org/CorpusID:120867665.
[15] MENON K V. Difference sets in Abelian groups[C/OL/1960. htps:/apisemanticscholar.orog/CorpusID:121354590.
[16] BAUMERT L D. Lecture Notes in Mathematics, 182: Cyclic difference sets[M].1st ed. 1971.ed. Berlin, Germany: Springer, 1971.
[17] LEHMER E. On Residue Difference Sets[/OL]. Canadian Jourmal of Mathematics, 1953, 5: 425 - 432. htpt:/api.scmantiscolar.on/CorusID:234799999
[18] BETH T, JUNGNICKEL D, LENZ H. Design Theory[M]. 2nd ed. Cambridge University Press, 1999.
[19] STANTON R G, SPROTTD A. A Family of Difference Sets[]. Canadian Journal of Mathematics, 1958, 10: 73-77.
[20] WHITEMAN A L. A Family of Difference Sets[J]. llinois Journal of Mathematics, 1962, 6 (1): 107-121.
[21] STORER T. Cyclotomy and difference sets[C/OL/1967. hts:/api.semanticholar.org/CorpusID:122519238.
[22] CAO Z. On Whiteman's and Storer's diference sets[J/OL]. Jourmal of Statistical Planning and Inference, 2001, 94: 147-154. htps:/apisemanticscholar.org/CorpusID:120449754.
[23] HALL M. Combinatorial Theory: HallCombinatorial[M]. Hoboken, N, USA: John Wiley & Sons, Inc., 1988.
[24] CALDERBANK R, KANTOR W M. The Geometry of Two-Weight Codes[J]. Bulletin of the London Mathematical Society, 1986, 18(2): 97-122.
[25] MOMIHARA K, XIANG Q. Strongly Regular Cayley Graphs from Partitions of Subdifference Sets of the Singer Difference Sets[]. Finite Fields and their Applications, 2018, 50: 222-250.
[26] BROUWER A E, HAEMERS W H. Spectra of Graphs[C/OL//2011. https://pisemanticscholaror/CorpusID:15634047.
[27] YAMAMOTO K. On congruences arising from relative Gauss sums[J]. Number Theory and Combinatorics, 1984: 423-446.
[28] MACWILLIAMS F, MANN H. On the p-rank of the design matrix of a difference set[J/OL]. Information and Contol, 1968, 12(5): 474- 488. https://www sciencedirect.com/science/article/pi/S0019995868905342. DOI: htp://oi.org/10.1016/S0019- 9958(68)90534-2.
[29] GOETHALS J M, DELSARTE P. On a class of majority-logic decodable cyclic codes[J/OL]. IEEE Trans. Inf. Theory, 1968, 14: 182-188. htps:/apisemanticschola org/CopusID:465861
[30] SMITH K. On the p-rank of the incidence matrix of points and hyperplanes in a finite projective geometry[J/OL]. Journal of Combinatorial Theory, 1969, 7(2): 122- 129. htp://www.sciencedirect.com/science/article/pii/S002 1980069800463. DOI: htps://doi org/10.1016/S0021-9800(69)80046- -3.
[31] EVANS R, HOLLMANN H, KRATTENTHALER C, et al. Gausss Sums, Jacobi Sums, and p-ranks of Cyclic Difference Sets[A]. 1998. arXiv: math/9807029.
[32] MCFARLAND R L. Sub-Difference Sets of Hadamard Difference Sets[]. Journal of Combinatorial Theory, Series A, 1990, 54(1): 112-122.
[33] MCELIECE R J. Ireducible Cyclic Codes and Gauss Sums[C]/HALL M, VAN LINT J H. Combinatorics. Dordrecht: Springer Netherlands, 1975: 185-202.
[34] GORDON B, MILLS W H, WELCH L R. Some New Difference Sets[/OL]. Canadian Journal of Mathematics, 1962, 14: 614- 625. http://apisermanticscholar.org/CorpusID:120892899.
[35] ARASU K T, HOLLMANN H D L, PLAYER K J, et al. ON THE p-RANKS OF GMW DIFFERENCE SETS[C/OL//2015. htpt:/pi.semanticscholar.org/CorpusID:123575905.
[36] GOLOMB s W. The Use of Combinatorial Structures in Communication Signal Decsign [M/OL]//Applications of Combinatorial Mathematics: Based on the proceedings of a conference on the Applications of Combinatorial Mathematics, organized by the Institute of Mathematics and its Applications and held at the University of Oxford in December 1994. Oxford University Press, 1997. htps:p//i.or/10./109/0/0/978019851 922.003.0005.
[37] SEGRE B. Ovals in a finite projective plane[J]. Canadian Journal of Mathematics, 1955, 7: 414-416.
[38] HIRSCHFELDJ W P. Projective geometries over finite fields[M]. Oxford University Press, 1998.
[39] GLYNN D. Two new sequences of ovals in finite Desarguesian planes of even order[J/OL]. Lecture Notes in Mathematics -Springer-verlag, 1983: 217-229. DOL: 10.1007/BFb0071521.
[40] MASCHIETTI A. Diference Sets and Hyperovals[J/OL]. Designs, Codes and Cryptography, 1998, 14: 89-98. htp://api.semanticscholar.org/CorpusID:9392408.
[41] DILLON J, DOBBERTIN H. New Cyclic Difference Sets with Singer Parameters[]. Finite Fields and Their Applications, 2004, 10(3): 342-389.
[42] NO J s, CHUNG H, YUN M. Binary Pseudorandom Sequences of Period 2m-1 with Ideal Autocorrelation Generated by the Polynomial zd + (z+1)d[J/OL]. IEEE Trans. Inf. Theory, 1998, 44: 1278-1282. htps://api.scmanticscholar.org/CorpusID:32641934.
[43] COLBOURN C J, DINITZ J H. Handbook of Combinatorial Designs[C/OL]/2006. https://api.semanticscholar.org/CorpusID:58219831.
[44] GAAL P, GOLOMB S W. Exhaustive determination of (1023, 511, 255)-cyclic difference sets[J/OL]. Math. Comput, 2001, 70: 357-366. htps:/apisematiccholar.or/CorpusID:10960299.
[45] ARASU K T, PLAYER K J. A New Family of Cyclic Difference Scts with Singer Parameters in Characteristic Three[J/OL]. Designs, Codes and Cryptography, 2003, 28: 75-91. https://apisemntischolar. org/CorpusID:21169693.
[46] HELLESETH T, KUMAR V, MARTINSEN H. A new family of termary sequences with idealtwo-level autocorrelation function[C/OL//2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060). 2000: 328-. DOI: 10.1 109/IT.200.866626.