Constructions of (q, K, λ, t, Q) almost difference families

Abstract

The concept of a (q, k, λ, t) almost difference family (ADF) has been introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, (q, K, λ, t, Q)-ADFs, where K = {k₁, k₂, ..., k_r} is a set of positive integers and Q = (q₁, q₂,..., q_r) is a given block-size distribution sequence. A necessary condition for the existence of a (q, K, λ, t, Q)-ADF is given, and several infinite classes of (q, K, λ, t, Q)-ADFs are constructed.

Keywords

Almost difference family (ADF) / cyclotomic class / difference family

References

[1] ArasuK T, DingC, HellesethT, KumarP V, MartinsenH. Almost difference sets and their sequences with optimal autocorrelation. IEEE Trans Inform Theory, 2001, 47: 2934-2943

[2] BrandstaetterN, WinterhofA. Some notes on the two-prime generator of order 2. IEEE Trans Inform Theory, 2005, 51: 3654-3657

[3] BurattiM. Recursive constructions for difference matrices and relative difference families. J Combin Des, 1998, 6: 165-182

[4] BurattiM. Pairwise balanced designs from finite fields. Discrete Math, 1999, 208/209: 103-107

[5] BurattiM. Cyclic designs with block size 4 and related optimal optical orthogonal codes. Des Codes Cryptogr, 2002, 26: 111-125

[6] BurattiM, PasottiA. Combinatorial designs and the theorem of Weil on multiplicative character sums. Finite Fields Appl, 2009, 15: 332-344

[7] BurattiM, WeiY, WuD, FanP, ChengM. Relative difference families with variable block sizes and their related OOCs. IEEE Trans Inform Theory, 2011, 57: 7489-7497

[8] CarletC, DingC. Highly nonlinear mappings. J Complexity, 2004, 20: 205-244

[9] ChangY, JiL. Optimal (4up, 5, 1) optical orthogonal codes. J Combin Des, 2004, 12: 346-361

[10] ChengM, MoZ, WuD. A note on balanced (q, {3, 4}, λ)-DFs with q a prime power. Australas J Combin, 2011, 50: 201-205

[11] ChungF R K, SalehiJ A, WeiV K. Optical orthogonal codes: design, analysis, and application. IEEE Trans Inform Theory, 1989, 35: 595-604

[12] ColbournC J, DinitzJ H, StinsonD R. Applications of combinatorial designs to communications, cryptography, and networking. In: LambJ D, PreeceD A, eds. Surveys in Combinatorics. Lond Math Soc, Lecture Note Ser, Vol 267. Cambridge: Cambridge Univ Press, 1999, 37-100

[13] DavisJ A. Almost difference sets and reversible difference sets. Arch Math, 1992, 59: 595-602

[14] DingC. Autocorrelation values of the generalized cyclotomic sequences of order 2. IEEE Trans Inform Theory, 1998, 44: 1699-1702

[15] DingC, YinJ. Constructions of almost difference families. Discrete Math, 2008, 308: 4941-4954

[16] LempelA, CohnM, EastmanW L. A class of binary sequences with optimal autocorrelation properties. IEEE Trans Inform Theory, 1977, 23: 38-42

[17] SidelnikovV M. Some k-valued pseudo-random sequences and nearly equidistant codes. Probl Inf Transm, 1969, 5: 12-16

[18] WangX, WangJ. A note on cyclic almost difference families. Discrete Math, 2011, 311: 628-633

[19] WangX, WuD. The existence of almost difference families. J Statist Plann Inference, 2009, 139: 4211-4216

[20] WangX, WuD, ChengM. The existence of (q, k, λ, t)-ADFs for k = 4, 5, 6. J Statist Plann Inference, 2010, 140: 3243-3251

[21] WilsonR M. Cyclotomic and difference families in elementary abelian groups. J Number Theory, 1972, 4: 17-47

[22] WuD, ChenZ, ChengM. A note on the existence of balanced (q, {3, 4}, 1) difference families. Australas J Combin, 2008, 41: 171-174

[23] YinJ. Some combinatorial constructions for optical orthogonal codes. Discrete Math, 1998, 185: 201-219

[24] ZhangY, LeiJ, ZhangS. A new family of almost difference sets and some necessary conditions. IEEE Trans Inform Theory, 2006, 52: 2052-2061

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