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https://digitalcommons.mtu.edu/math-fp
Recent documents in Department of Mathematical Sciences Publicationsen-usWed, 08 Nov 2023 13:57:23 PST3600A new bound on the number of designs with classical affine parameters
https://digitalcommons.mtu.edu/math-fp/174
https://digitalcommons.mtu.edu/math-fp/174Tue, 11 Jun 2019 07:04:29 PDTClement Lam et al.Extremal self-dual codes from symmetric designs
https://digitalcommons.mtu.edu/math-fp/172
https://digitalcommons.mtu.edu/math-fp/172Tue, 11 Jun 2019 06:56:46 PDT
A method is given for constructing extremal binary self-dual [42, 21, 8]-codes from certain symmetric 2-(41, 16, 6) designs.
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Edward Spence et al.Concerning multiplier automorphisms of cyclic Steiner triple systems
https://digitalcommons.mtu.edu/math-fp/171
https://digitalcommons.mtu.edu/math-fp/171Tue, 11 Jun 2019 06:52:10 PDT
A cyclic Steiner triple system, presented additively over Z_{v} as a set B of starter blocks, has a non-trivial multiplier automorphism λ ≠ 1 when λB is a set of starter blocks for the same Steiner triple system. When does a cyclic Steiner triple system of order v having a nontrivial multiplier automorphism exist? Constructions are developed for such systems; of most interest, a novel extension of Netto's classical construction for prime orders congruent to 1 (mod 6) to prime powers is proved. Nonexistence results are then established, particularly in the cases when v = (2^{β} + 1)^{α}, when v = 9p with p ≡ 5 (mod 6), and in certain cases when all prime divisors are congruent to 5 (mod 6). Finally, a complete solution is given for all v < 1000, in which the remaining cases are produced by simple computations.
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Charles Colbourn et al.On Kirkman triple systems of order 33
https://digitalcommons.mtu.edu/math-fp/170
https://digitalcommons.mtu.edu/math-fp/170Tue, 11 Jun 2019 06:40:36 PDT
Twenty-eight non-isomorphic KTS(33) with an automorphism of order 11 are constructed from the 84 cyclic STS(33).
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Vladimir Tonchev et al.Some small non-embeddable designs
https://digitalcommons.mtu.edu/math-fp/169
https://digitalcommons.mtu.edu/math-fp/169Tue, 11 Jun 2019 06:34:20 PDT
Examples of non-embeddable 2-(15, 7, 6) and 2-(16, 8, 7) designs are constructed.
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Vladimir TonchevQuasi-symmetric designs, codes, quadrics, and hyperplane sections
https://digitalcommons.mtu.edu/math-fp/168
https://digitalcommons.mtu.edu/math-fp/168Thu, 06 Jun 2019 12:12:26 PDT
It is proved that a quasi-symmetric design with the Symmetric Difference Property (SDP) is uniquely embeddable as a derived or a residual design into a symmetric SDP design. Alternatively, any quasi-symmetric SDP design is characterized as the design formed by the minimum weight vectors in a binary code spanned by the simplex code and the incidence vector of a point set in PG(2m-1, 2) that intersects every hyperplane in one of two prescribed numbers of points. Applications of these results for the classification of point sets in PG(2m-1, 2) with the same intersection properties as an elliptic or a hyperbolic quadric, as well as the classification of codes achieving the Grey-Rankin bound are discussed.
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Vladimir TonchevOn the extendability of steiner t‐designs
https://digitalcommons.mtu.edu/math-fp/167
https://digitalcommons.mtu.edu/math-fp/167Thu, 06 Jun 2019 11:59:06 PDT
Necessary and sufficient conditions for the extendability of residual designs of Steiner systems S(t,t + 1,v) are studied. In particular, it is shown that a residual design with respect to a single point is uniquely extendable, and the extendability of a residual design with respect to a pair of points is equivalent to a bipartition of the block graph of a related design.
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Alphonse Baartmans et al.A symmetric 2‐(160, 54, 18) design
https://digitalcommons.mtu.edu/math-fp/166
https://digitalcommons.mtu.edu/math-fp/166Thu, 06 Jun 2019 11:53:29 PDT
We give a construction for a symmetric 2‐(160, 54, 18) design that effectively uses the incidence matrix of the design of the hyperplanes in PG(3, 3) as a tactical decomposition. This is the first design ever found with these parameters.
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Edward Spence et al.A design and a code invariant under the simple group Co3
https://digitalcommons.mtu.edu/math-fp/165
https://digitalcommons.mtu.edu/math-fp/165Thu, 06 Jun 2019 10:25:06 PDT
A self-orthogonal doubly-even (276, 23) code invariant under Conway's simple group Co_{3} is constructed. The minimum weight codewords form a 2-(276, 100, 1458) doubly transitive block-primitive design with block stabilizer isomorphic to the Higman-Sims simple group HS. More generally, the codewords of any given weight are single orbits stabilized by maximal subgroups of Co_{3}. The restriction of the code on the complement of a minimum weight codeword is the (176, 22) code discovered by Calderbank and Wales as a code invariant under HS.
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Willem Haemers et al.Designs with the symmetric difference property on 64 points and their groups
https://digitalcommons.mtu.edu/math-fp/164
https://digitalcommons.mtu.edu/math-fp/164Thu, 06 Jun 2019 10:17:16 PDT
The automorphism groups of the symmetric 2-(64, 28, 12) designs with the symmetric difference property (SDP), as well as the groups of their derived and residual designs, are computed. The symmetric SDP designs all have transitive automorphism groups. In addition, they all admit transitive regular subgroups, or equivalently, (64, 28, 12) difference sets. These results are used for the enumeration of certain binary codes achieving the Grey-Rankin bound and point sets of elliptic or hyperbolic type in PG(5, 2).
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Christopher Parker et al.The preparata codes and a class of 4‐designs
https://digitalcommons.mtu.edu/math-fp/163
https://digitalcommons.mtu.edu/math-fp/163Thu, 06 Jun 2019 10:10:05 PDT
An extension theorem for t‐designs is proved. As an application, a class of 4‐(4^{m} + 1,5,2) designs is constructed by extending designs related to the 3‐designs formed by the minimum weight vectors in the Preparata code of length n = 4^{m}, m ≥ 2.
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Alphonse Baartmans et al.Linear codes and doubly transitive symmetric designs
https://digitalcommons.mtu.edu/math-fp/162
https://digitalcommons.mtu.edu/math-fp/162Thu, 06 Jun 2019 08:07:46 PDT
An application of linear codes for the construction of a 2-transitive symmetric design from its residual or derived designs is discussed. The Higman design is constructed as the design supported by the minimum weight codewords in the extended binary code of certain designs invariant under PΣU (3, 5^{2}).
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Christopher Parker et al.Singly-even self-dual codes and Hadamard matrices
https://digitalcommons.mtu.edu/math-fp/161
https://digitalcommons.mtu.edu/math-fp/161Thu, 06 Jun 2019 08:00:56 PDT
A construction of binary self-dual singly-even codes from Hadamard matrices is described. As an application, all inequivalent extremal singly-even [40,20,8] codes derived from Hadamard matrices of order 20 are enumerated.
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Masaaki Harada et al.The existence of extremal self-dual [50,25,10] codes and quasi-symmetric 2-(49,9,6) designs
https://digitalcommons.mtu.edu/math-fp/160
https://digitalcommons.mtu.edu/math-fp/160Thu, 06 Jun 2019 07:52:14 PDT
All extremal binary self-dual [50,25,10] codes with an automorphism of order 7 are enumerated. Up to equivalence, there are four such codes, three with full automorphism group of order 21, and one code with full group of order 7. The minimum weight codewords yield quasi-symmetric 2-(49,9,6) designs.
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W. Cary Huffman et al.On quasi-symmetric 2-(28,12,11) and 2-(36,16,12) designs
https://digitalcommons.mtu.edu/math-fp/159
https://digitalcommons.mtu.edu/math-fp/159Thu, 06 Jun 2019 07:45:41 PDT
New quasi-symmetric 2-(28,12,11) and 2-(36,16,12) designs are constructed by embedding known designs into symmetric designs.
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Clement Lam et al.Classification of affine resolvable 2-(27, 9, 4) designs
https://digitalcommons.mtu.edu/math-fp/158
https://digitalcommons.mtu.edu/math-fp/158Thu, 06 Jun 2019 07:30:19 PDT
All affine resolvable designs with parameters of the design of the hyperplanes in ternary affine 3-space are enumerated. This enumeration implies the classification (up to equivalence), of all optimal equidistant ternary codes of length 13 and distance 9, as well as all complete orthogonal arrays of strength 2 with 3 symbols, 13 constraints and index 3. Up to isomorphism, there are exactly 68 such designs. The automorphism groups and the rank of the incidence matrices over GF(3) are computed. There are six designs with point-transitive automorphism groups, and one design with trivial group. The affine geometry design is the unique design with lowest 3-rank, and the only design with 2-transitive automorphism group.
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Clement Lam et al.The existence of certain extremal [54,27,10] self-dual codes
https://digitalcommons.mtu.edu/math-fp/157
https://digitalcommons.mtu.edu/math-fp/157Thu, 06 Jun 2019 07:25:22 PDT
Some new extremal binary [54,27,10] self-dual codes are constructed using automorphisms of order 7.
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Vladimir Tonchev et al.Spreads in strongly regular graphs
https://digitalcommons.mtu.edu/math-fp/156
https://digitalcommons.mtu.edu/math-fp/156Thu, 06 Jun 2019 07:15:58 PDT
A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte's bound (also called Huffman's bound). Such spreads give rise to colorings meeting Hoffman's lower bound for the chromatic number and to certain imprimitive three-class association schemes. These correspondences lead to conditions for existence. Most examples come from spreads and fans in (partial) geometries. We give other examples, including a spread in the McLaughlin graph. For strongly regular graphs related to regular two-graphs, spreads give lower bounds for the number of non-isomorphic strongly regular graphs in the switching class of the regular two-graph.
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Willem Haemers et al.On the binary codes of Steiner triple systems
https://digitalcommons.mtu.edu/math-fp/155
https://digitalcommons.mtu.edu/math-fp/155Thu, 06 Jun 2019 07:09:21 PDT
The binary code spanned by the rows of the point by block incidence matrix of a Steiner triple system STS(ν) is studied. A sufficient condition for such a code to contain a unique equivalence class of STS(ν)'s of maximal rank within the code is proved. The code of the classical Steiner triple system defined by the lines in PG(n-1, 2)(n (≥) 3, or AG(n, 3(n ≥) 3 is shown to contain exactlyν codewords of weight r = (ν 1)/2, hence the system is characterized by its code. In addition, the code of the projective STS(2 n -1) is characterized as the unique (up to equivalence) binary linear code with the given parameters and weight distribution. In general, the number of STS(ν)'s contained in the code depends on the geometry of the codewords of weightr. It is demonstrated that the ovals and hyperovals of the defining STS(ν) play a crucial role in this geometry. This relation is utilized for the construction of some infinite classes of Steiner triple systems without ovals.
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Alphonse Baartmans et al.The uniformly packed binary [27,21,3] and [35,29,3] codes☆
https://digitalcommons.mtu.edu/math-fp/154
https://digitalcommons.mtu.edu/math-fp/154Thu, 06 Jun 2019 06:57:12 PDT
All binary projective two-weight codes with parameters [27,6,12] and [35,6,16] are enumerated up to equivalence, and the automorphism groups of the codes and related strongly regular graphs computed. Alternatively, this implies the enumeration of all linear binary uniformly packed codes with parameters [27,21,3] and [35,29,3] as dual codes of two-weight codes.
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Vladimir Tonchev