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New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

Jong-Seon No
2004 Designs, Codes and Cryptography  
In this paper, for a prime power q, new cyclic difference sets with Singer parameters ððq n À 1=q À 1Þ; ðq nÀ1 À 1=q À 1Þ; ðq nÀ2 À 1=q À 1ÞÞ are constructed by using q-ary sequences (d-homogeneous functions  ...  When q is a power of 3, new cyclic difference sets with Singer parameters ððq n À 1=q À 1Þ; ðq nÀ1 À 1=q À 1Þ; ðq nÀ2 À 1=q À 1ÞÞ are constructed from the ternary sequences of period q n À 1 with ideal  ...  In this section, it is shown for a prime power q that cyclic difference sets with Singer parameter can be constructed from the d-homogeneous functions with difference-balance property.  ... 
doi:10.1023/b:desi.0000036246.52472.81 fatcat:elf64mv6nva4vnl43xbmtdkzae

New Cyclic Relative Difference Sets Constructed From$d$-Homogeneous Functions With Difference-Balanced Property

S.-H. Kim, J.-S. No, H. Chung, T. Helleseth
2005 IEEE Transactions on Information Theory  
In this paper, for any prime power q, it is shown that new cyclic relative difference sets with parameters ( q n −1 q−1 , q−1, q n−1 , q n−2 ) can be constructed by using d-homogeneous functions on F q  ...  Several new cyclic relative difference sets with parameters ( q n −1 q−1 , q − 1, q n−1 , q n−2 ) are constructed by using p-ary sequences of period q n − 1 with ideal autocorrelation property introduced  ...  No introduced the construction method of a new cyclic difference set with Singer parameters from the d-homogeneous function with difference-balanced property [8] .  ... 
doi:10.1109/tit.2004.842712 fatcat:6bc4azywpvewjcvsnj54ne36ja

Difference Balanced Functions and Their Generalized Difference Sets [article]

Alexander Pott, Qi Wang
2013 arXiv   pre-print
In the literature, all existing such functions are associated with the d-homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be d-homogeneous.  ...  First we characterize difference balanced functions by generalized difference sets with respect to two exceptional subgroups.  ...  Furthermore, the projection ρ(D b ) is a cyclic difference set with Singer parameters p n −1 p−1 , p n−1 , p n−2 (p − 1) for each b ∈ F * p and ρ(D 0 ) is the complement of ρ(D b ), where ρ : F * p n →  ... 
arXiv:1309.7842v1 fatcat:ozpyylng6ffh3nujq4cqam7b2y

Recent progress in algebraic design theory

Qing Xiang
2005 Finite Fields and Their Applications  
We survey recent results on difference sets, p-ranks and Smith normal forms of certain set-inclusion matrices and subspace-inclusion matrices.  ...  No [96] used d-homogeneous functions on F * q m over F q with difference-balanced property to construct cyclic difference sets with classical parameters.  ...  Difference sets with classical parameters The Singer difference sets arise from the classical designs of points and hyperplanes in projective space PG(m − 1, q); they are cyclic difference sets with parameters  ... 
doi:10.1016/j.ffa.2005.06.003 fatcat:5v7swzvqpreijnuzd6z6uagulm

Page 430 of Mathematical Reviews Vol. , Issue 85b [page]

1985 Mathematical Reviews  
At the same time, the authors construct over 500000 new, simple, nontrivial 5-designs with v = 33, by using the 4-homogeneous group PI'L2(32). Sanpe: Kageyama (Hiroshima) 85b:05033 85b:05034  ...  A Singer design is a symmetric block design with parameters (qr +q™-' + ---+qt1,q™-) + ---+q+1,q""? + ---+94+1),¢ a prime power.  ... 

Formal solutions of differential equations

Michael. F Singer
1990 Journal of symbolic computation  
By using the term "formal", I wish to emphasize the fact that most of the time I will not be concerned with questions of where power series converge or in what domains the expressions represent functions  ...  Consider the linear differential equation z(y) = an_l(x)y/n-1) + . . . + a0(x)Y = 0 This is an e x p a n d e d and revised version of talks presented at  ...  For example, C(x) with the derivation d/dx in a differential field as is the field of meromorphic functions on a connected open set in C with the usual derivation.  ... 
doi:10.1016/s0747-7171(08)80037-5 fatcat:rz54tv4j3jezxjxugn22mr4kji

Gauss Sums, Jacobi Sums, and p-ranks of Cyclic Difference Sets [article]

Ronald Evans, Henk Hollmann, Christian Krattenthaler
1998 arXiv   pre-print
We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2^d-1, 2^d-1-1, 2^d-2-1) cyclic difference sets in the multiplicative  ...  We determine the 2-ranks of the difference sets arising from the Segre and Glynn hyperovals, in the following way.  ...  The research of the fourth was supported in part by NSA grant MDA 904-97-0104, and a grant from the University of Delaware Research Foundation.  ... 
arXiv:math/9807029v2 fatcat:vs5em2tsdrb2pbwnqz42yv22v4

Gauss Sums, Jacobi Sums, and p-Ranks of Cyclic Difference Sets

Ronald Evans, Henk D.L. Hollmann, Christian Krattenthaler, Qing Xiang
1999 Journal of combinatorial theory. Series A  
We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2 d − 1, 2 d−1 − 1, 2 d−2 − 1) cyclic difference sets in the finite  ...  We determine the 2-ranks of the difference sets arising from the Segre and Glynn hyperovals, in the following way.  ...  The research of the fourth was supported in part by NSA grant MDA 904-97-0104, and a grant from the University of Delaware Research Foundation.  ... 
doi:10.1006/jcta.1998.2950 fatcat:jkx5jmvmofc2rpe2myci3y2b3u

Orbit codes from forms on vector spaces over a finite field

Angela Aguglia, ,Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Bari, I-70126, Italy, Antonio Cossidente, Giuseppe Marino, Francesco Pavese, Alessandro Siciliano, ,Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Potenza, I-85100, Italy, ,Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli "Federico II", Napoli, I-80138, Italy
2019 Advances in Mathematics of Communications  
In this paper we construct different families of orbit codes in the vector spaces of the symmetric bilinear forms, quadratic forms and Hermitian forms on an n-dimensional vector space over the finite field  ...  In this paper we aim at establishing new techniques for the construction of orbit codes.  ...  Note that for k ≤ n − k, starting from a suitable set of k × (n − k)-matrices with entries in F q , known as MRD-codes, by means of the so-called lifting process, one can construct an (n, q (n−k)(k−d+1  ... 
doi:10.3934/amc.2020105 fatcat:udkkhljsxvh6fkcgbinp5i546m

On Rational Solutions of Systems of Linear Differential Equations

Moulay A. Barkatou
1999 Journal of symbolic computation  
We propose a new algorithm to compute the set of rational solutions of such a system. This algorithm does not require the use of cyclic vectors.  ...  We show how one can use this algorithm to give a method to find the set of solutions with entries in K(x)[log x] of M(Y ) = N .  ...  One then obtains a new set of parameters P, a set C of linear constraints on these parameters and a polynomial sol = x µ c µ where the c µ 's are column vectors, the components of which are linear in the  ... 
doi:10.1006/jsco.1999.0314 fatcat:xqpgeyuhnvctdbl7vciwmrljsu

Ring geometries, Two-Weight Codes and Strongly Regular Graphs [article]

E. Byrne, M. Greferath, T. Honold
2007 arXiv   pre-print
These examples all yield infinite families of strongly regular graphs with non-trivial parameters.  ...  Here we extend this classical result to the case of a ring-linear code with exactly two nonzero homogeneous weights and multisets of points in an associated projective ring geometry.  ...  These point sets provide new constructions of non-trivial strongly regular graphs.  ... 
arXiv:0709.0862v1 fatcat:umhmpehjdjcgrdtvg7bdrhxl2u

Ring geometries, two-weight codes, and strongly regular graphs

Eimear Byrne, Marcus Greferath, Thomas Honold
2008 Designs, Codes and Cryptography  
These examples all yield infinite families of strongly regular graphs with non-trivial parameters.  ...  Here we extend this classical result to the case of a ring-linear code with exactly two nonzero homogeneous weights and multisets of points in an associated projective ring geometry.  ...  These point sets provide new constructions of non-trivial strongly regular graphs.  ... 
doi:10.1007/s10623-007-9136-8 fatcat:3i7b3d6zhrfwxbhywmmleadq6q

Further combinatorial constructions for optimal frequency-hopping sequences

Gennian Ge, Ryoh Fuji-Hara, Ying Miao
2006 Journal of combinatorial theory. Series A  
Theory 50 (2004) 2408-2420] from a combinatorial approach, where a correspondence between frequency-hopping (FH) sequences and partition-type cyclic difference packings was established, and several combinatorial  ...  constructions were provided for FHMA systems with a single optimal FH sequence.  ...  Zhu at Suzhou University for their helpful discussions on the constructions in Sections 4 and 6, respectively, and the two anonymous referees for their helpful comments.  ... 
doi:10.1016/j.jcta.2006.03.019 fatcat:663deipeabgjnapcuxrksjysfe

The family index theorem and bifurcation of solutions of nonlinear elliptic bvp [article]

Jacobo Pejsachowicz
2012 arXiv   pre-print
The results are of different nature from the ones that can be obtained via the traditional Lyapunov-Schmidt reduction.  ...  Our sufficient conditions for bifurcation are derived from the Atiyah-Singer family index theorem and therefore they depend only on the coefficients of derivatives of leading order of the linearized differential  ...  Then the approximations are constructed using the fact that the set of restrictions to the unit sphere of elements of A(n, r) = ∪ k≥0 A(n, r, 2k) is dense in the set of even continuos functions from the  ... 
arXiv:1107.1727v3 fatcat:2dppb3c52bevtjyjjcycwbg3di

Linear Codes from Some 2-Designs [article]

Cunsheng Ding
2015 arXiv   pre-print
In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets  ...  A classical method of constructing a linear code over (q) with a t-design is to use the incidence matrix of the t-design as a generator matrix over (q) of the code.  ...  Sihem Mesnager, for their constructive comments that much improved the presentation and quality of this paper.  ... 
arXiv:1503.06511v1 fatcat:4c2m5zgge5g33bnol32vdxsst4
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