Department of Mathematical Sciences Publications
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Recent documents in Department of Mathematical Sciences Publications
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Sun, 12 May 2019 02:46:58 PDT
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Selforthogonal codes from symmetric designs with fixedpointfree automorphisms
https://digitalcommons.mtu.edu/mathfp/126
https://digitalcommons.mtu.edu/mathfp/126
Fri, 10 May 2019 12:48:43 PDT
In this paper, we consider a method for constructing nonbinary selforthogonal codes from symmetric designs with fixedpointfree automorphisms. All codes over GF(3) and GF(7) derived from symmetric 2(v,k,λ) designs with fixedpointfree automorphisms of order p for the parameters (v,k,λ,p)=(27,14,7,3),(40,27,18,5) and (45,12,3,5) are classified. A ternary [63,20,21] code with a record breaking minimum weight is constructed from the symmetric 2(189,48,12) design found recently by Janko. Several codes over GF(5) and GF(7) that are either optimal or have the largest known minimum weight are constructed from designs obtained from known difference sets.
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Masaaki Harada et al.

A formula for the number of Steiner quadruple systems on 2n points of 2‐rank 2n−n
https://digitalcommons.mtu.edu/mathfp/125
https://digitalcommons.mtu.edu/mathfp/125
Fri, 10 May 2019 12:34:47 PDT
Assmus [1] gave a description of the binary code spanned by the blocks of a Steiner triple or quadruple system according to the 2‐rank of the incidence matrix. Using this description, the author [13] found a formula for the total number of distinct Steiner triple systems on 2^{n}−1 points of 2‐rank 2^{n} ‐n. In this paper, a similar formula is found for the number of Steiner quadruple systems on 2^{n} points of 2‐rank 2^{n} ‐n. The formula can be used for deriving bounds on the number of pairwise non‐isomorphic systems for large n, and for the classification of all non‐isomorphic systems of small orders. The formula implies that the number of non‐isomorphic Steiner quadruple systems on 2^{n} points of 2‐rank 2^{n} ‐n grows exponentially. As an application, the Steiner quadruple systems on 16 points of 2‐rank 12 are classified up to isomorphism. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 260–274, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10036
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Vladimir Tonchev

Numerical solution of spacetime fractional PDEs using RBFQR and Chebyshev polynomials
https://digitalcommons.mtu.edu/mathfp/124
https://digitalcommons.mtu.edu/mathfp/124
Tue, 07 May 2019 13:57:50 PDT
In this study, we propose a numerical discretization of spacetime fractional partial differential equations (PDEs) with variable coefficients, based on the radial basis functions (RBF) and pseudospectral (PS) methods. The RBF method is used for space discretization, while Chebyshev polynomials handle time discretization. The use of PS methods significantly reduces the number of nodes needed to obtain the solution. The proposed numerical scheme is capable of handling all three types of boundary conditions: Dirichlet, Neumann and Robin. We give numerical examples to validate our method and to show its superior performance compared to other techniques.
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Sushil Kumar et al.

A note on MDS codes, nArcs and complete designs
https://digitalcommons.mtu.edu/mathfp/123
https://digitalcommons.mtu.edu/mathfp/123
Fri, 03 May 2019 10:44:46 PDT
A generalized incidence matrix of a design over GF(q) is any matrix obtained from the (0, 1)incidence matrix by replacing ones with nonzero elements from GF(q). The dimension d q of a design D over GF(q) is defined as the minimum value of the qrank of a generalized incidence matrix of D. It is proved that the dimension d_{q} of the complete design on n points having as blocks all wsubsets, is greater that or equal to n − w + 1, and the equality d_{q} = n − w + 1 holds if and only if there exists an [n, n − w + 1, w] MDS code over GF(q), or equivalently, an narc in PG(w − 2, q).
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Vladimir Tonchev

A new quasisymmetric 2(56,16,6) design obtained from codes
https://digitalcommons.mtu.edu/mathfp/122
https://digitalcommons.mtu.edu/mathfp/122
Fri, 03 May 2019 10:38:20 PDT
The binary code spanned by the blocks of the known quasisymmetric 2(56,16,6) design is utilized for the construction of a new quasisymmetric design with these parameters. The new design is then embedded as a residual design into a new nonselfdual symmetric 2(78,22,6) design.
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Akihiro Munemasa et al.

On generalized Hadamard matrices of minimum rank
https://digitalcommons.mtu.edu/mathfp/121
https://digitalcommons.mtu.edu/mathfp/121
Fri, 03 May 2019 10:25:23 PDT
Generalized Hadamard matrices of order q^{n−1} (q—a prime power, n⩾2) over GF(q) are related to symmetric nets in affine 2(q^{n},q^{n−1},(q^{n−1}−1)/(q−1)) designs invariant under an elementary abelian group of order q acting semiregularly on points and blocks. The rank of any such matrix over GF(q) is greater than or equal to n−1. It is proved that a matrix of minimum qrank is unique up to a monomial equivalence, and the related symmetric net is a classical net in the ndimensional affine geometry AG(n,q).
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Vladimir Tonchev

A characterization of designs related to an extremal doublyeven selfdual code of length 48
https://digitalcommons.mtu.edu/mathfp/120
https://digitalcommons.mtu.edu/mathfp/120
Fri, 03 May 2019 10:15:15 PDT
The uniqueness of a binary doublyeven selfdual [48, 24, 12] code is used to prove that a selforthogonal 5(48, 12, 8) design, as well as some of its derived and residual designs, including a quasisymmetric 2(45, 9, 8) design, are all unique up to isomorphism.
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Masaaki Harada et al.

Symmetric (4,4)Nets and Generalized Hadamard Matrices Over Groups of Order 4
https://digitalcommons.mtu.edu/mathfp/119
https://digitalcommons.mtu.edu/mathfp/119
Thu, 02 May 2019 08:56:47 PDT
The symmetric classregular (4,4)nets having a group of bitranslations G of order four are enumerated up to isomorphism. There are 226 nets with G≅Z_{2}×Z_{2} , and 13 nets with G≅Z_{4} . Using a (4,4)net with full automorphism group of smallest order, the lower bound on the number of pairwise nonisomorphic affine 2(64,16,5) designs is improved to 21,621,600. The classification of classregular (4,4)nets implies the classification of all generalized Hadamard matrices (or difference matrices) of order 16 over a group of order four up to monomial equivalence. The binary linear codes spanned by the incidence matrices of the nets, as well as the quaternary and Z_{4} codes spanned by the generalized Hadamard matrices are computed and classified. The binary codes include the affine geometry [64,16,16] code spanned by the planes in AG(3,4) and two other inequivalent codes with the same weight distribution.These codes support nonisomorphic affine 2(64,16,5) designs that have the same 2rank as the classical affine design in AG(3,4), hence provide counterexamples to Hamada’s conjecture. Many of the F_{4} codes spanned by generalized Hadamard matrices are selforthogonal with respect to the Hermitian inner product and yield quantum errorcorrecting codes, including some codes with optimal parameters.
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Masaaki Harada et al.

Affine designs and linear orthogonal arrays☆
https://digitalcommons.mtu.edu/mathfp/118
https://digitalcommons.mtu.edu/mathfp/118
Thu, 02 May 2019 08:48:17 PDT
It is proved that the collection of blocks of an affine 1design that yields a linear orthogonal array is a union of parallel classes of hyperplanes in a finite affine space. In particular, for every prime power q and every m⩾2 there exists a unique (up to equivalence) complete linear orthogonal array of strength two associated with the classical design of points and hyperplanes in AG(m,q).
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Vladimir Tonchev

Partitions of difference sets and code synchronization
https://digitalcommons.mtu.edu/mathfp/117
https://digitalcommons.mtu.edu/mathfp/117
Thu, 02 May 2019 08:39:25 PDT
Difference systems of sets (DSS) are combinatorial structures that are a generalization of cyclic difference sets and arise in connection with code synchronization. The paper surveys recent constructions and open problems concerning DSS obtained as partitions of cyclic difference sets.
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Vladimir Tonchev

Hyperplane partitions and difference systems of sets☆
https://digitalcommons.mtu.edu/mathfp/116
https://digitalcommons.mtu.edu/mathfp/116
Thu, 02 May 2019 08:33:31 PDT
Difference Systems of Sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper gives new constructions of DSS obtained from partitions of hyperplanes in a finite projective space, as well as DSS obtained from balanced generalized weighing matrices and partitions of the complement of a hyperplane in a finite projective space.
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Ryoh FujiHara et al.

On affine designs and GMW difference sets
https://digitalcommons.mtu.edu/mathfp/115
https://digitalcommons.mtu.edu/mathfp/115
Wed, 01 May 2019 11:14:02 PDT
A bound on the 2rank of an affine designs obtained from a Hadamard design with a line spread via Rahilly's construction is proved and applied to Hadamard designs related to difference sets of GMW type. Some questions motivated by the recent discovery of new counterexamples to the conjectures of A... and Hamada about characterizing geometric designs in terms of their rank are discussed.
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Vladimir Tonchev

An algorithm for optimal difference systems of sets
https://digitalcommons.mtu.edu/mathfp/114
https://digitalcommons.mtu.edu/mathfp/114
Wed, 01 May 2019 10:40:48 PDT
Difference Systems of Sets (DSS) are combinatorial structures that generalize cyclic difference sets and are used in code synchronization. A DSS is optimal if the associated code has minimum redundancy for the given block length n, alphabet size q, and errorcorrecting capacity ρ. An algorithm for finding optimal DSS is presented together with tables of optimal solutions found by this algorithm.
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Vladimir Tonchev et al.

On conflictavoiding codes of length n=4m for the active users
https://digitalcommons.mtu.edu/mathfp/113
https://digitalcommons.mtu.edu/mathfp/113
Wed, 01 May 2019 10:34:25 PDT
New improved upper and lower bounds on the maximum size of a symmetric or arbitrary conflictavoiding code of length n = 4 m for three active users are proved. Furthermore, direct constructions for optimal conflictavoiding codes of length n = 4 m and m equiv 2 (mod 4) for three active users are provided.
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Masakazu Jimbo et al.

A Class of 2(3n7, 3n−1 7, (3n−1 7−1)/2) Designs
https://digitalcommons.mtu.edu/mathfp/112
https://digitalcommons.mtu.edu/mathfp/112
Wed, 01 May 2019 10:18:11 PDT
Generalized Hadamard matrices are used for the construction of a class of quasi‐residual nonresolvable BIBD's with parameters . The designs are not embeddable as residual designs into symmetric designs if n is even. The construction yields many nonisomorphic designs for every given n ≥ 2, including more than 10^{17 }nonisomorphic 2‐(63,21,10) designs.
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Vladimir Tonchev

Difference systems of sets and cyclotomy
https://digitalcommons.mtu.edu/mathfp/111
https://digitalcommons.mtu.edu/mathfp/111
Wed, 01 May 2019 08:57:27 PDT
Difference systems of sets (DSS) are combinatorial configurations that arise in connection with code synchronization. A method for the construction of DSS from partitions of cyclic difference sets was introduced in [V.D. Tonchev, Difference systems of sets and code synchronization, Rend. Sem. Mat. Messina, Ser. II, t. XXV 9 (2003) 217–226] and applied to cyclic difference sets (n, (n−1)/2, (n−3)/4) of Paley type, where n ≡ 3(mod 4) is a prime number. This paper develops similar constructions for prime numbers n ≡ 1(mod 4) that use partitions of the set of quadratic residues, as well as more general cyclotomic classes.
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Yukiyasu Mutoh et al.

On affine designs and Hadamard designs with line spreads☆
https://digitalcommons.mtu.edu/mathfp/110
https://digitalcommons.mtu.edu/mathfp/110
Wed, 01 May 2019 08:50:36 PDT
Rahilly [On the line structure of designs, Discrete Math. 92 (1991) 291–303] described a construction that relates any Hadamard design H on 4^{m} − 1 points with a line spread to an affine design having the same parameters as the classical design of points and hyperplanes in AG(m, 4). Here it is proved that the affine design is the classical design of points and hyperplanes in AG(m, 4) if, and only if, H is the classical design of points and hyperplanes in PG(2m−1, 2) and the line spread is of a special type. Computational results about line spreads in PG(5, 2) are given. One of the affine designs obtained has the same 2rank as the design of points and planes in AG(3, 4), and provides a counterexample to a conjecture of Hamada [On the prank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to errorcorrecting codes, Hiroshima Math. J. 3 (1973) 153–226].
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V. C. Mavron et al.

Cyclic quasisymmetric designs and selforthogonal codes of length 63☆
https://digitalcommons.mtu.edu/mathfp/109
https://digitalcommons.mtu.edu/mathfp/109
Wed, 01 May 2019 08:41:42 PDT
The enumeration of binary cyclic selforthogonal codes of length 63 is used to prove that any cyclic quasisymmetric 2(63, 15, 35) design with block intersection numbers x=3 and y=7 is isomorphic to the geometric design having as blocks the threedimensional subspaces in PG(5, 2).
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Chekad Sarami et al.

Steiner systems for twostage disjunctive testing
https://digitalcommons.mtu.edu/mathfp/108
https://digitalcommons.mtu.edu/mathfp/108
Wed, 01 May 2019 08:34:36 PDT
The subject of this paper are some constructions of Steiner designs with blocks of two sizes that differ by one. The study of such designs is motivated by a combinatorial lower bound on the minimum number of individual tests at the second stage of a 2stage disjunctive testing algorithm.
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Vladimir Tonchev

Quantum codes from caps
https://digitalcommons.mtu.edu/mathfp/107
https://digitalcommons.mtu.edu/mathfp/107
Wed, 01 May 2019 08:28:49 PDT
Caps in a finite projective geometry over GF(4) are used for the construction of some quantum errorcorrecting codes, including an optimal [[ 27; 13; 5 ]] code.
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Vladimir Tonchev