arXiv subjects

Dragomir Z. Djokovic

Publications and source records attributed to Dragomir Z. Djokovic.

At least 55 records · Page 3Linked to original sources

arXiv2010

Classification of base sequences BS(n+1,n)

Base sequences BS(n+1,n) are quadruples of {1,-1}-sequences (A;B;C;D), with A and B of length n+1 and C and D of length n, such that the sum of their nonperiodic autocorrelation functions is a delta-function. The base sequence conjecture, asserting that BS(n+1,n) exist for all n, is stronger than the famous Hadamard matrix conjecture. We introduce a new definition of equivalence for base sequences BS(n+1,n) and construct a canonical form. By using this canonical form, we have enumerated the equivalence classes of BS(n+1,n) for n <= 30. Due to excessive size of the equivalence classes, the tables in the paper cover only the cases n <= 13.

math.CO↗

arXiv2010

On the base sequence conjecture

Let BS(m,n) denote the set of base sequences (A;B;C;D), with A and B of length m and C and D of length n. The base sequence conjecture (BSC) asserts that BS(n+1,n) exist (i.e., are non-empty) for all n. This is known to be true for n <= 36 and when n is a Golay number. We show that it is also true for n=37 and n=38. It is worth pointing out that BSC is stronger than the famous Hadamard matrix conjecture. In order to demonstrate the abundance of base sequences, we have previously attached to BS(n+1,n) a graph Gamma_n and computed the Gamma_n for n <= 27. We now extend these computations and determine the Gamma_n for n=28,...,35. We also propose a conjecture describing these graphs in general.

math.CO↗

arXiv2010

Hadamard matrices from base sequences: An example

There are several well-known methods that one can use to construct Hadamard matrices from base sequences BS(m,n). In view of the recent classification of base sequences BS(n+1,n) for n <= 30, it may be of interest to show on an example how prolific these methods are. For that purpose we have selected the Hadamard matrices of order 60. By using these methods and the transposition map we have constructed 1759 nonequivalent Hadamard matrices of order 60.

math.CO↗

arXiv2010

Some new near-normal sequences

The normal sequences NS(n) and near-normal sequences NN(n) play an important role in the construction of orthogonal designs and Hadamard matrices. They can be identified with certain base sequences (A;B;C;D), where A and B have length n+1 and C and D length n. C.H. Yang conjectured that near-normal sequences exist for all even n. While this has been confirmed for n not exceeding 30, so far nothing else was known for larger n. We show that NN(32) consists of 8 equivalence classes and we exhibit their representatives. We also construct representatives for two equivalence classes of NN(34). On the other hand our exhaustive computer searches show that NS(31) and NS(33) are void.

math.CO↗

arXiv2009

Hadamard matrices of small order and Yang conjecture

We show that 138 odd values of n less than 10000 for which one knows how to construct a Hadamard matrix of order 4n have been overlooked in the recent handbook of combinatorial designs. There are four additional odd n, namely 191, 5767, 7081 and 8249, in that range for which we can construct a Hadamard matrix of order 4n. Our exhaustive computer searches show that the near-normal sequences NN(n) exist for n=36,38,40. Thus the Yang conjecture on the existence of NN(n) for all even n has been verified for n <= 40 but it still remains open.

math.CO↗

arXiv2009

Zero patterns and unitary similarity

A subspace of the space, L(n), of traceless complex $n\times n$ matrices can be specified by requiring that the entries at some positions $(i,j)$ be zero. The set, $I$, of these positions is a (zero) pattern and the corresponding subspace of L(n) is denoted by $L_I(n)$. A pattern $I$ is universal if every matrix in L(n) is unitarily similar to some matrix in $L_I(n)$. The problem of describing the universal patterns is raised, solved in full for $n\le3$, and partial results obtained for $n=4$. Two infinite families of universal patterns are constructed. They give two analogues of Schur's triangularization theorem.

math.RT↗

arXiv2009

Supplementary difference sets with symmetry for Hadamard matrices

First we give an overview of the known supplementary difference sets (SDS) (A_i), i=1..4, with parameters (n;k_i;d), where k_i=|A_i| and each A_i is either symmetric or skew and k_1 + ... + k_4 = n + d. Five new Williamson matrices over the elementary abelian groups of order 25, 27 and 49 are constructed. New examples of skew Hadamard matrices of order 4n for n=47,61,127 are presented. The last of these is obtained from a (127,57,76)-difference family that we have constructed. An old non-published example of G-matrices of order 37 is also included.

math.CO↗

arXiv2009

Universal subspaces for compact Lie groups

For a representation of a connected compact Lie group G in a finite dimensional real vector space U and a subspace V of U, invariant under a maximal torus of G, we obtain a sufficient condition for V to meet all G-orbits in U, which is also necessary in certain cases. The proof makes use of the cohomology of flag manifolds and the invariant theory of Weyl groups. Then we apply our condition to the conjugation representations of U(n), Sp(n), and SO(n) in the space of $n\times n$ matrices over C, H, and R, respectively. In particular, we obtain an interesting generalization of Schur's triangularization theorem.

math.RT↗

arXiv2009

Classification of near-normal sequences

We introduce a canonical form for near-normal sequences NN(n), and using it we enumerate the equivalence classes of such sequences for even n up to 30. These sequences are needed for Yang multiplication in the construction of longer T-sequences from base sequences.

math.CO↗

arXiv2009

Cyclic (v;r,s;lambda) difference families with two base blocks and v <= 50

We construct many new cyclic (v;r,s;lambda) difference families with v less than or equal 50. In particular we construct the difference families with parameters (45;18,10;9), (45;22,22;21), (47;21,12;12), (47;19,15;12), (47;22,14;14), (48;20,10;10), (48;24,4;12), (50;25,20;20) for which the existence question was an open problem. The (45;22,22;21) difference family gives a BIBD with parameters v=45, b=90, r=44, k=22 and lambda=21, and the one with parameters (50;25,20;20) gives a pair of binary sequences of length 50 with zero periodic autocorrelation function (the periodic analog of a Golay pair). We also construct nine new D-optimal designs. A normal form for cyclic difference families is proposed and used effectively in compiling the list of known and new difference families.

math.CO↗

arXiv2009

Rational Orthogonal versus Real Orthogonal

The main question we raise here is the following one: given a real orthogonal n by n matrix X, is it true that there exists a rational orthogonal matrix Y having the same zero-pattern? We conjecture that this is the case and prove it for n<=5. We also consider the related problem for symmetric orthogonal matrices.

math.CO↗

arXiv2009

On polynomial invariants of several qubits

It is a recent observation that entanglement classification for qubits is closely related to local $SL(2,\CC)$-invariants including the invariance under qubit permutations, which has been termed $SL^*$ invariance. In order to single out the $SL^*$ invariants, we analyze the $SL(2,\CC)$-invariants of four resp. five qubits and decompose them into irreducible modules for the symmetric group $S_4$ resp. $S_5$ of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of $SL^*$-invariants vanishing on arbitrary product states. We find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced in [Phys. Rev. A 72, 012337 (2005)]. Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the $Ω$-process. As the degrees of invariants increase, the alternative method proves to be particularly efficient.

quant-ph↗

arXiv2008

Periodic complementary sets of binary sequences

Let PCS_p^N denote a set of p binary sequences of length N such that the sum of their periodic auto-correlation functions is a delta-function. In the 1990, Boemer and Antweiler addressed the problem of constructing such sequences. They presented a table covering the range p <= 12, N <= 50 and showing in which cases it was known at that time whether such sequences exist, do not exist, or the question of existence is undecided. The number of undecided cases was rather large. Subsequently the number of undecided cases was reduced to 26 by the author. In the present note, several cyclic difference families are constructed and used to obtain new sets of periodic binary sequences. Thereby the original problem of Boemer and Antweiler is completely solved.

cs.IT↗

arXiv2008

Skew-Hadamard matrices of orders 188 and 388 exist

We construct several difference families on cyclic groups of orders 47 and 97, and use them to construct skew-Hadamard matrices of orders 188 and 388. Such difference families and matrices are constructed here for the first time. The matrices are constructed by using the Goethals-Seidel array.

math.CO↗

arXiv2008

On orthogonal and special orthogonal invariants of a single matrix of small order

We study the structure of the algebra of polynomial invariants for the usual conjugation action of the complex special, SO_n, and general, O_n, orthogonal group on the space of traceless n by n complex matrices. (Note that these two algebras coincide if n is odd.) Minimal generating sets of these algebras are known for n less than 5. We construct one for n=5. We also construct a Hironaka decomposition in the case n=3 and a new (more economical) such decomposition for n=4. A simple presentation (with just one syzygy) is obtained for the algebra in the case n=3.

math.AC↗

arXiv2007

Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities

We construct six unitary trace invariants for 2 by 2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We prove two quaternionic versions of a well known characterization of triangularizable subalgebras of matrix algebras over an algebraically closed field. Finally we consider the problem of describing the semi-algebraic set of pairs (X,Y) of quaternionic n by n matrices which are simultaneously triangularizable. Even the case n=2, which we analyze in more detail, remains unsolved.

math.AC↗

arXiv2007

On orthostochastic, unistochastic and qustochastic matrices

We introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm. This is the quaternionic analogue of the unistochastic matrices studied by physicists. We also introduce quaternionic Hadamard matrices and quaternionic mutually unbiased bases (MUB). In particular we show that the number of MUB in an n-dimensional quaternionic Hilbert space is at most 2n+1. The bound is attained for n=2. We also determine all quaternionic Hadamard matrices of size at most 4.

math-ph↗

arXiv2007

Skew-Hadamard matrices of orders 436, 580 and 988 exist

We construct two difference families on each of the cyclic groups of order 109, 145 and 247, and use them to construct skew-Hadamard matrices of orders 436, 580 and 988. Such difference families and matrices are constructed here for the first time. The matrices are constructed by using the Goethals-Seidel array.

math.CO↗