Uniri projekt / UNIRI project

Kodovi, grupe i kombinatoričke strukture / Codes, groups and combinatoral structures

Odjel za matematiku Sveučilišta u Rijeci / Department of Mathematics, University of Rijeka

 

Odjel za matematiku

Sveučilište u Rijeci

 

 

U sklopu ovog projekta konstruirat će se i analizirati različiti tipovi kombinatoričkih dizajna i grafova, kao i kodovi određeni njihovim matricama incidencije i matricama susjedstva, odnosno orbitnim matricama. Kombinatoričke strukture konstruirat će se kombiniranjem algebarskih i geometrijskih metoda te uz primjenu računala. Proučavat će se i drugi kombinatorički objekti, kao što su binarni komplementarni nizovi (npr. periodični Golayjevi parovi) i Hadamardove matrice (uključujući kompleksne Hadamardove matrice). Također će se proučavati veza s drugim strukturama, na primjer s konačnim geometrijama i asocijacijskim shemama. Konačne grupe imat će značajnu ulogu u konstrukciji i analizi kombinatoričkih struktura. Kao rezultate predloženog istraživanja očekujemo konstrukcije i klasifikacije različitih tipova kombinatoričkih dizajna i grafova (npr. novih blokovnih dizajna i jako regularnih, odnosno distance-regularnih, grafova) i drugih kombinatoričkih objekata (komplementarnih nizova, Hadamardovih matrica, itd.), kao i konstrukciju kodova s dobrim svojstvima (npr. linearnih kodova s velikom minimalnom udaljenošću s obzirom na duljinu i dimenziju koda, samoortogonalnih odnosno samodualnih kodova, itd.). Očekujemo da će rezultati ovog projekta biti zanimljivi znanstvenicima koji rade u području teorije dizajna, teorije kodiranja i teorije grafova, kao i onima koji se bave istraživanjima u području teorije konačnih grupa.

This project deals with construction and analysis of various types of combinatorial designs and graphs, as well as codes determined by their incidence or adjacency matrices, or their orbit matrices. For construction of combinatorial structures we will combine computational, geometric and algebraic approach. We will also take into consideration other combinatorial objects, such as binary complementary sequences (e.g. periodic Golay pairs) and Hadamard matrices (including complex Hadamard matrices). Relations to other structures (finite geometries, association schemes, etc.) will also be taken into account. Finite groups actions will have a significant role in constructing and analysing combinatorial structures. As a result of the proposed research, we expect a construction and classification of various types of combinatorial designs and graphs (e.g. new block designs and strongly/distance regular graphs) and other combinatorial objects (complementary sequences, Hadamard matrices, etc.), as well as a construction of codes with good properties (e.g. linear codes with large minimum distance comparing to the length and the dimension of the code, self-orthogonal or self-dual codes, etc.). We expect that the outcomes of this project will be of interest for researchers working in design theory, coding theory and graph theory, and for those working in theory of finite groups.

 

Voditelj / Principal investigator:

prof. dr. sc. Dean Crnković (Odjel za matematiku Sveučilišta u Rijeci / Department of Mathematics, University of Rijeka ), e-mail: deanc@math.uniri.hr

Istraživački tim / Research team:

Ana Grbac (Odjel za matematiku Sveučilišta u Rijeci / Department of Mathematics, University of Rijeka)

dr. sc. Daniel Hawtin (Odjel za matematiku Sveučilišta u Rijeci / Department of Mathematics, University of Rijeka)

doc. dr. sc. Marija Maksimović (Odjel za matematiku Sveučilišta u Rijeci / Department of Mathematics, University of Rijeka)

dr. sc. Nina Mostarac (Odjel za matematiku Sveučilišta u Rijeci / Department of Mathematics, University of Rijeka)

doc. dr. sc. Andrea Švob (Odjel za matematiku Sveučilišta u Rijeci / Department of Mathematics, University of Rijeka)

Tin Zrinski (Odjel za matematiku Sveučilišta u Rijeci / Department of Mathematics, University of Rijeka)

 

Objavljeni radovi / Papers:

  • R. F. Bailey, D. R. Hawtin, On the 486-vertex distance-regular graphs of Koolen-Riebeek and Soicher, Electron. J. Combin. 27 (2020), #P3.13, 12 pages.

  • D. Crnković, R. Egan, A. Švob, Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs, Adv. Math. Commun. 14 (2020), 591-602.

  • D. Crnković, M. Maksimović, Strongly regular graphs with parameters (37,18,8,9) having nontrivial automorphisms, Art Discrete Appl. Math. 3 (2020), #P2.10, 8 pages.

  • J. D'haeseleer, J. Mannaert, L. Storme, A. Švob, Cameron-Liebler line classes in AG(3,q), Finite Fields Appl. 67 (2020) 101706, 17 pages.

  • D. Crnković, S. Rukavina, A. Švob, On some distance-regular graphs with many vertices, J. Algebraic Combin. 51 (2020), 641-652.

  • D. Crnković, M. Maksimović, Construction of strongly regular graphs having an automorphism group of composite order, Contrib. Discrete Math. 15 (2020), 22-41.

  • D. Crnković, F. Pavese, A. Švob, On the PSU(4,2)-invariant vertex-transitive strongly regular (216,40,4,8) graph, Graphs Combin. 36 (2020), 503-513.

  • D. Crnković, H. Kharaghani, A. Švob, Divisible design Cayley digraphs, Discrete Math. 343 (2020), 111784, 8 pages.

  • A. Švob, Transitive distance-regular graphs from linear groups L(3,q), q=2,3,4,5, Trans. Comb. 9 (2020), 49-60.

  • M. Maksimović, Self-orthogonal codes from row orbit matrices of strongly regular graphs, Sarajevo J. Math. 15 (2019), 309-322.

  • S. Ban, D. Crnković, M. Mravić, S. Rukavina, New extremal Type II Z4-codes of length 32 obtained from Hadamard matrices, Discrete Math. Algorithms Appl. 11 (2019), 1950057, 18 pages (electronic) (http://dx.doi.org/10.1142/S1793830919500575).

  • A. Švob, Transitive distance-regular graphs from the unitary groups U(3,4), U(4,3) and U(5,2), Bull. Inst. Combin. Appl. 87 (2019), 103-113.

  • D. Crnković, A. Švob, V. Tonchev, Cyclotomic trace codes, Algorithms, 12 (2019) 168, 10 pages (electronic) (https://doi.org/10.3390/a12080168).

  • M. De Boeck, M. Rodgers, L. Storme, A. Švob, Cameron-Liebler sets of generators in finite classical polar spaces, J. Combin. Theory Ser. A 167 (2019), 340-388.

  • D. Crnković, S. Rukavina, M. Šimac, LDPC codes from μ-geodetic graphs obtained from block designs, Graphs Combin. 35 (2019), 451-469.

  • D. Crnković, D. Dumičić Danilović, S. Rukavina, M. Šimac, On some new Steiner 2-designs S(2,5,45), Util. Math. 111 (2019), 281-308.

  • D. Crnković, A. Švob, Transitive t-designs and strongly regular graphs constructed from linear groups L(2,q), q <= 23, Int. J. Group Theory 8 (2019), 43-64.

  • D. Crnković, V. Mikulić Crnković, A. Švob, Block designs and strongly regular graphs admitting a transitive action of the Mathieu group M11, Australas. J. Combin. 73 (2019), 149-161.

  • D. Crnković, R. Egan, A. Švob, Constructing self-orthogonal and Hermitian self-orthogonal codes via weighing matrices and orbit matrices, Finite Fields Appl. 55 (2019), 64-77.

 

Pozvana predavanja / Invited talks:  

  • D. Crnković, New constructions of Deza digraphs, The 3rd Workshop on Algebraic Graph Theory and its Applications, Akademgorodok, Russia, November 2 - 8, 2020.

  • A. Švob, Strongly regular graphs with parameters (81,20,9,12) and a new partial geometry, The 3rd Workshop on Algebraic Graph Theory and its Applications, Akademgorodok, Russia, November 2 - 8, 2020

 

Izlaganja na konferencijama / Talks:  

  • D. Crnković, Self-orthogonal codes from block designs and association schemes, 9th Slovenian Conference on Graph Theory, Bled, Slovenia, June 23 - 29, 2019.

  • A. Švob, Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs, 9th Slovenian Conference on Graph Theory, Bled, Slovenia, June 23 - 29, 2019.

  • T. Zrinski, Constructing block designs from orbit matrices using a modified genetic algorithm, 9th PhD Summer School in Discrete Mathematics, Rogla, Slovenia, June 30 - July 5, 2019.

  • N. Mostarac, PD-sets for codes related to flag-transitive symmetric designs, Finite Geometry & Friends (a Brussels Summer School on Finite Geometry), Brussels, Belgium, June 17 - 21, 2019.


Doktorske disertacije / PhD theses:  

  • A. Grbac, Self-dual and LCD codes from two class association schemes, November 12, 2020.


Članovi istraživačkog tima izlažu o svojim rezultatima i u okviru Seminara za konačnu matematiku.

 

Odjel za matematiku Sveučilište u Rijeci