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46
Normal Bases over Finite Fields
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
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Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...
Almost Difference Sets and Their Sequences With Optimal Autocorrelation
, 2001
"... Almost difference sets have interesting applications in cryptography and coding theory. In this paper, we give a wellrounded treatment of known families of almost difference sets, establish relations between some difference sets and some almost difference sets, and determine the numerical multiplie ..."
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Cited by 15 (2 self)
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Almost difference sets have interesting applications in cryptography and coding theory. In this paper, we give a wellrounded treatment of known families 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 H @�� � RA 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 R with optimal autocorrelation.
Cyclotomic constructions of skew Hadamard difference sets
 J. Combin. Theory (A
"... (This is a sample cover image for this issue. The actual cover is not yet available at this time.) This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors i ..."
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Cited by 12 (6 self)
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(This is a sample cover image for this issue. The actual cover is not yet available at this time.) This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are
An Experimental Search and New Combinatorial Designs via a Generalisation of Cyclotomy
 J. Combin. Math. Combin. Comput
, 1997
"... Cyclotomy can be used to construct a variety of combinatorial designs, for example, supplementary difference sets, weighing matrices and T matrices. These designs may be obtained by using linear combinations of the incidence matrices of the cyclotomic cosets. However, cyclotomy only works in the p ..."
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Cited by 11 (6 self)
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Cyclotomy can be used to construct a variety of combinatorial designs, for example, supplementary difference sets, weighing matrices and T matrices. These designs may be obtained by using linear combinations of the incidence matrices of the cyclotomic cosets. However, cyclotomy only works in the prime and prime power cases. We present a generalisation of cyclotomy and introduce generalised cosets. Combinatorial designs can now be obtained by a search through all linear combinations of the incidence matrices of the generalised cosets. We believe that this search method is new. The generalisation works for all cases and is not restricted to prime powers. The paper presents some new combinatorial designs. We give a new construction for T matrices of order 87 and hence an OD(4 \Theta 87; 87; 87; 87; 87). We also give some Doptimal designs of order n = 2v = 2 \Theta 145; 2 \Theta 157; 2 \Theta 181. Keywords: Cyclotomy, Galois field, Galois domain, autocorrelation function, supplemen...
PROOFS OF TWO CONJECTURES ON TERNARY WEAKLY REGULAR BENT FUNCTIONS
"... We study ternary monomial functions of the form f(x) = Trn(ax d), where x ∈ F3n and Trn: F3n → F3 is the absolute trace function. Using a lemma of Hou [17], Stickelberger’s theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising from ..."
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Cited by 8 (5 self)
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We study ternary monomial functions of the form f(x) = Trn(ax d), where x ∈ F3n and Trn: F3n → F3 is the absolute trace function. Using a lemma of Hou [17], Stickelberger’s theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising from [12] are weakly regular bent, settling a conjecture of Helleseth and Kholosha [12]. We also prove that the CoulterMatthews bent functions are weakly regular.
Modified group divisible designs with block size 4
"... Abstract: It is shown here that the necessary conditions for the existence of MGD[4, A m, n] for A ~ 2 are sufficient with the exception of MGO(4, 3, 6,23]. 1. ..."
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Abstract: It is shown here that the necessary conditions for the existence of MGD[4, A m, n] for A ~ 2 are sufficient with the exception of MGO(4, 3, 6,23]. 1.
Table of the existence of Hadamard matrices
, 1990
"... On Hadamard matrices Recent advances in the construction of Hadamard matrices have depended on the existence of BaumertHall arrays and four (1,1) matrices A, B, C, D of order m which are of Williamson type, that is pairwise satisfy (i) MNT = NMT and (ii) AAT + BBT + CCT + DDT = 4mlm. If (i) is rep ..."
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Cited by 6 (2 self)
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On Hadamard matrices Recent advances in the construction of Hadamard matrices have depended on the existence of BaumertHall arrays and four (1,1) matrices A, B, C, D of order m which are of Williamson type, that is pairwise satisfy (i) MNT = NMT and (ii) AAT + BBT + CCT + DDT = 4mlm. If (i) is replaced by (i')MN = NM we have GoethalsSeidel matrices. These matrices are very important to the determination of the Hadamard conjecture: that there exists an Hadamard matrix of order 4t for all natural numbers t. This paper shows how the Williamson type and GoethalsSeidel type Hadamard matrices may be combined by introducing Fmatrices which are a generalization of both Williamson and GoethalsSeidel matrices. Several constructions for Fmatrices are given showing they exist for the new orders 119, 171, 185, 217 and the new classes 1/4q(q + I), q = 3(mod 8) a prime power and 1/2p(p 3), p 4(mod 4) and p 4 both prime powers (among others).
Jacobi sums and new families of irreducible polynomials of Gaussian periods
 Math. Comp
"... Abstract. Let m>2, ζm an mth primitive root of 1, q ≡ 1mod2m a prime number, s = sq a primitive root modulo q and f = fq =(q − 1)/m. We study the Jacobi sums Ja,b = − ∑ q−1 k=2 ζ a inds(k)+b inds(1−k) m,0 ≤ a, b ≤ m − 1, where inds(k) is the least nonnegative integer such that s inds(k) ≡ k m ..."
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Abstract. Let m>2, ζm an mth primitive root of 1, q ≡ 1mod2m a prime number, s = sq a primitive root modulo q and f = fq =(q − 1)/m. We study the Jacobi sums Ja,b = − ∑ q−1 k=2 ζ a inds(k)+b inds(1−k) m,0 ≤ a, b ≤ m − 1, where inds(k) is the least nonnegative integer such that s inds(k) ≡ k mod q. We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families Pq(x), q ∈P, of irreducible polynomials of Gaussian periods, ηi = ∑f−1 q,ofdegreem, wherePis j=0 ζsi+mj a suitable set of primes ≡ 1mod2m. We exhibit examples of such families for several small values of m, and give a MAPLE program to construct more of them.
Hadamard Matrices, Orthogonal Designs and Construction Algorithms
"... We discuss algorithms for the construction of Hadamard matrices. We include discussion of construction using Williamson matrices, Legendre pairs and the discret Fourier transform and the two circulants construction. Next we move to algorithms to determine the equivalence of Hadamard matrices using t ..."
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Cited by 5 (1 self)
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We discuss algorithms for the construction of Hadamard matrices. We include discussion of construction using Williamson matrices, Legendre pairs and the discret Fourier transform and the two circulants construction. Next we move to algorithms to determine the equivalence of Hadamard matrices using the pro le and projections of Hadamard matrices. A summary is then given which considers inequivalence of Hadamard matrices of orders up to 44. The nal two sections give algorithms for constructing orthogonal designs, short amicable and amicable sets for use in the Kharaghani array. 1 Algorithms for constructing Hadamard matrices 1.1 Hadamard matrices constructed from Williamson matrices An Hadamard matrix H of order n has elements 1 and satis es HH T = nI n . These matrices are used extensively in coding and communications (see Seberry and Yamada [90]). The order of an Hadamard matrix is 1, 2 or n (0 mod 4). The rst unsolved case is order 428. We use Williamson's construction as the basis of our algorithm to construct a distributed computer search for new Hadamard matrices. We briey describe the theory of Williamson's construction below. Previous computer searches for Hadamard matrices using Williamson's condition 2 are described in Section 1.1.1. The implementation of the search algorithm is presented in Section 1.1.2, and the results of the search are described in Section 1.1.3. Theorem 1 (Williamson [104]) Suppose there exist four (1; 1) matrices A, B, C, D of order n which satisfy XY T = Y X T ; X;Y 2 fA; B; C; Dg Further, suppose AA T +BB T + CC T +DD T = 4nI n (1) Then H = 2 6 6 6 4 A B C D B A D C C D A B D C B A 3 7 7 7 5 (2) is an Hadamard matrix of order 4n constructed from a Williamson array. Let the matrix T given below be called ...