Curriculum of Luca Giuzzi
Personal details
Surname:  Giuzzi 
Name:  Luca 
Title:  Dr. (DPhil, Sussex) 
Date of Birth:  30 September 1973 
Citizenship:  Italian 
Known languages:  Italian (mothertongue), 
English (fluent).  
Telephone (BS):  +39 030 3715739 
Fax:  +39 030 3715745 
Email:  luca.giuzzi@unibs.it 
Web page:  http://lucagiuzzi.unibs.it 
ORCID:  0000000339757281 
ResearcherID:  F40662010 
Address (work):  DICATAM 
Section of Mathematics  
Facoltà di Ingegneria  
Università degli studi di Brescia  
via Branze 43  
25123 Brescia  
ITALY  
Address (home):  via Cesare Battisti 42 
25018 Montichiari (BS)  
ITALY  
Studies
 Laurea in matematica (degree in mathematics) with a score of 110 cum laude/110, Università Cattolica del Sacro Cuore; thesis "Gruppi di Frobenius e strutture geometriche associate" under the supervision of Prof. Silvia Pianta, 1996.
 DPhil in Mathematics, University of Sussex; thesis "Hermitian varieties over finite field" under the supervision of Prof. James Hirschfeld, 2001.
Employment history
 19921996: Undergraduate student at Università Cattolica del Sacro Cuore.
 19961997: Unix system administrator at Università Cattolica.
 19972000: DPhil student at University of Sussex.
 20002004: research fellow at Università degli studi di Brescia.
 20022003: work on "Development of a cryptosystem based on elliptic or hyperelliptic curves" (POP 342) in collaboration with Università della Basilicata.
 2004April 2008: research associate in Geometry at Politecnico di Bari.
 May 2008now: research associate in Geometry at Università degli Studi di Brescia.
Awards and grants
 May 1997: grant for `advanced studies abroad' by Università Cattolica/
 July 1997: research grant for `doctoral studies' by Istituto Nazionale di Alta Matematica; this grant has been renewed for the years 19981999 and 19992000.
 September 19972000: British EPSRC award (covering tuition fees at Sussex University).
 June 2000: research grant for `Progetto e realizzazione di un criptosistema per comunicazioni" (Project and implementation of a cryptosystem for communication systems), POPFESR 1994/99, Università della Basilicata.
 September 2000December 2004: research fellowship at Università degli studi di Brescia.
 2001: Exhonerated from compulsory military service as "Citizen involved with special merits in activities of national or international relevance in science, art or culture", Art. 7(d) DLG 504/97.
 2015, "Incentivo una tantum anno 2011" (competitive award from the from the University of Brescia for activities in the years 20082011).
Conference organization
 2007: Member of the organizing committee of the "XVIII Congresso UMI", Bari.
 2008: Organizer of the international conference "Combinatorics 2008", Brescia.
 2017: Organizer of the international conference "Fq13", Gaeta.
 2017: Organizer of the international conference "HyGraDe", S. Alessio Siculo (Messina).
 2018: Organizer of the international conference "Combinatorics 2018".
Involvment with National Research Projects (PRIN)
 2001: "Strutture geometriche, combinatorica e loro applicazioni", as member;
 2003: "Strutture geometriche, combinatoria e loro applicazioni", as member;
 2005: "Grafi e Geometrie" (2005018845\_002), as member;
 2008: "Strutture di incidenza e combinatorie" (2008BHF4AW\_003), as member;
 2012: "Strutture geometriche, combinatoria e loro applicazioni," (2012XZE22K\_009) as Scientific Coordinator of the local research unit.
European Projects
 Participant to COST action IC1104: "Random Network Coding and Designs over $\mathrm{GF}(q)$"
Teaching experience
Dr. Giuzzi has been a professore aggregato since September 2006. 19981999: teaching assistant for Discrete Mathematics at University of Sussex.
 19992000: teaching assistant for Discrete Mathematics and Group Theory at University of Sussex.
 20002001: minicourse (10 hours) on cryptography at Università Cattolica del Sacro Cuore.
 20012002: minicourse on "Cryptography: from DES to AES", Università di Brescia.
 20032004: course (10 hours) on "Discrete Mathematics", Università di Brescia.
 20042005: main instructor for "Geometria Superiore 2", Università Cattolica del Sacro Cuore.
 20042005: teaching assistant at Politecnico di Bari (3 courses).
 September 2005: introductory course on geometry at Politecnico di Bari (10 hours).
 20052006: teaching assistant at Politecnico di Bari (3 courses).
 20062007: main instructor for "Geometria e Algebra", Mechanical Engineering, A, Politecnico di Bari.
 20072008: main instructor for "Geometria e Algebra", Mechanical Engineering, A, Politecnico di Bari.
 2007: seminars (3 hours) on Coding Theory at Università degli studi di Brescia, within the European Social Found project n. 411498 Ob. 3 Mis. C3 (2006), 414809, no. 671794 "Corso di alta formazione nell'ambito di dottorati afferenti al settore dell'Ingegneria dell'informazione" (Higher education course for doctoral studies related to information engineering).
 2008: cosupervisor for first and second level degree theses in Mathematics at Università Cattolica.
 20082010: main instructor for "Complementi di Geometria e Algebra", Università degli Studi di Brescia.
 20102011: main instructor for "Algebra per Codici e Crittografia" (Algebra for codes and cryptography), Università degli Studi di Brescia.

20112012: main instructor for
 "Algebra per Codici e Crittografia", Università degli Studi di Brescia;
 "Geometria combinatorica" (Combinatorial geometry), Università Cattolica del S. Cuore (BS).

20122013: main instructor for
 "Algebra per Codici e Crittografia", Università degli Studi di Brescia;
 "Algebra e Geometria", Università degli studi di Brescia.

20132014: main instructor for
 "Algebra per Codici e Crittografia" ($6$ CFU), Università degli Studi di Brescia;
 "Algebra e Geometria" ($9$ CFU), Università degli studi di Brescia.

20142015: main instructor for
 "Algebra per Codici e Crittografia" ($6$ CFU), Università degli Studi di Brescia;
 "Algebra e Geometria" ($9$ CFU), Università degli studi di Brescia.

20152016: main instructor for
 "Algebra per Codici e Crittografia" ($6$ CFU), Università degli Studi di Brescia;
 "Algebra e Geometria" ($9$ CFU), Università degli studi di Brescia.

20162017: main instructor for
 "Algebra per Codici e Crittografia" ($6$ CFU), Università degli Studi di Brescia;
 "Algebra e Geometria" ($9$ CFU), Università degli studi di Brescia.

20172018: main instructor for
 "Algebra per Codici e Crittografia" ($6$ CFU), Università degli Studi di Brescia;
 "Algebra e Geometria" ($9$ CFU), Università degli studi di Brescia.
Accepted papers
 L. Giuzzi, "Collineation groups of the intersection of two classical unitals", J. Comb. Des. 9: 445–459 (2001) ISSN: 10638539, doi: 10.1002/jcd.1023.
 L. Giuzzi, H. Karzel, "CoMinkowski spaces, their reflection structure and Kloops", Discrete Math. 255: 161–179 (2002), ISSN: 0012365X, doi: 10.1016/S0012365X(01)00396X.
 L. Giuzzi, "A characterisation of classical unitals", J. Geom., 74: 86–89 (2002), ISSN: 00472468, doi: 10.1007/PL00012541.
 L. Giuzzi, G. Korchmáros, "Ovoids of the Hermitian Surface in Odd Characteristic", Adv. Geom., Special Issue (2003), S49–S58, ISSN: 1615715X.
 L. Giuzzi, "On the intersection of Hermitian surfaces", J. Geom., 85: 49–60 (2006), ISSN: 00472468, doi: 10.1007/s0002200600424.
 L. Giuzzi, "A geometric construction for some ovoids of the Hermitian Surface", Results Math. 49: 81–88 (2006), ISSN: 14226383, doi: 10.1007/s0002500602108.
 A. Aguglia, L. Giuzzi, "Orthogonal arrays from Hermitian varieties", Innov. Incidence Geom. 5: 129–144 (2007), ISSN: 17816475 (arxiv: 0705.3590).
 A. Aguglia, L. Giuzzi, "Construction of a $3$–dimensional MDS code", Contrib. Discrete Math. 3 (1), 39–46 (2007), ISSN: 17150868, doi: 10.1007/s000250070268y (arxiv: 0708.1558).
 A. Aguglia, L. Giuzzi, "An algorithm for constructing some maximal arcs in $\mathrm{PG}(2,q^2)$", Results Math. 52 no. 1–2: 17–33 (2008), ISSN: 14226383, doi: 10.1007/s000250070268y (arxiv: math/0611466).
 A. Aguglia, L. Giuzzi, G. Korchmáros, "Algebraic curves and maximal arcs", J. Algebraic Combin. 28: 531–544 (2008), ISSN: 09259899, doi: 10.1007/s1080100801227 (arxiv: math/0702770).
 A. Aguglia, L. Giuzzi, "On the non–existence of certain hyperovals in dual André planes of order $2^{2k}$", Electron. J. Combin. 15(1): N37 (2008); (arxiv: 0803.1597).
 L. Giuzzi, A. Sonnino, "LDPC codes from Singer cycles", Discrete Appl. Math. 157: 1723–1728 (2009), ISSN: 0166218X, doi: 10.1016/j.dam.2009.01.013 (arxiv: 0709.2813).
 A. Aguglia, L. Giuzzi, G. Korchmáros, "Construction of unitals in Desarguesian planes", Discrete Math. 310 (22): 3162–3167 (2010), ISSN: 0012365X, doi: 10.1016/j.disc.2009.06.023 (arxiv: 0810.2233).
 L. Giuzzi, A. Pasotti, "Sampling complete graphs", Discrete Math. 312 (3), 488–497 (2012), ISSN: 0012365X, doi: 10.1016/j.disc.2011.02.034 (arxiv: 0907.3199).
 L. Giuzzi, G. Korchmáros, "Unitals in $PG(2,q^2)$ with a large $2$point stabiliser, Discrete Math. 312 (3): 532–535 (2012), ISSN: 0012365X doi: 10.1016/j.disc.2011.03.017 (arxiv: 1009.6109).
 A. Benini, L. Giuzzi, A. Pasotti, "Downlinking $(K_v,\Gamma)$–designs to $P_3$–designs", Util. Math. 90: 3–21 (2013) ISSN: 03153681 (arxiv: 1004.4127).
 A. Benini, L. Giuzzi, A. Pasotti, "New results on pathdecompositions and their downlinks", Util. Math. 90: 369–382 (2013) ISSN: 03153681 (arxiv: 1106.1095).
 L. Giuzzi, V. Pepe, "Families of twisted tensor product codes", Des. Codes Cryptogr. 67: 375–384 (2013) ISSN: 09251022 doi: 10.1007/s1062301296136 (arxiv: 1107.1066).
 I. Cardinali, L. Giuzzi, "Codes and caps from orthogonal Grassmannians ", Finite Fields Appl. 24: 148–169 (2013) ISSN: 10715797 doi: 10.1016/j.ffa.2013.07.003 (arxiv: 1303.5636).
 A. Aguglia, L. Giuzzi, "Intersections of the Hermitian surface with irreducible quadrics in $PG(3,q^2)$, $q$ odd", Finite Fields Appl. 30: 1–13 (2014) ISSN: 10715797 doi: 10.1016/j.ffa.2014.05.005 (arxiv: 1307.8386).
 L. Giuzzi, V. Pepe, " On some subvarieties of the Grassmann variety", Linear Multilinear Algebra 63 (11): 2121–2134 (2015) ISSN: 03081087 doi: 10.1080/03081087.2014.983449 (arxiv: 1405.6926).
 I. Cardinali, L. Giuzzi, "Minimum distance of Symplectic Grassmann Codes", Linear Algebra Appl. 488: 124–134 (2016) ISSN: 00243795 doi: 10.1016/j.laa.2015.09.031 (arxiv: 1503.05456).
 I. Cardinali, L. Giuzzi, K.V. Kaipa, A. Pasini, "Line Polar Grassmann Codes of Orthogonal Type", J. Pure Appl. Algebra 220 (5): 19241934 (2016) ISSN: 00224049 doi: 10.1016/j.jpaa.2015.10.007.
 A. Aguglia, L. Giuzzi, "Intersections of the Hermitian Surface with irreducible Quadrics in even Characteristic", Electron. J. Combin. 23 (4): P4.13 (2016) (arxiv: 1407.8498).
 A. Aguglia, L. Giuzzi, "Intersection sets, threecharacter multisets and associated codes", Des. Codes Cryptogr. 83: 269282 (2017) doi: 10.1007/s1062301603028 (arxiv: 1504.00503).
 I. Cardinali, L. Giuzzi, A. Pasini, "A geometric approach to alternating $k$linear forms", J. Algebraic Combin. 45: 931963 (2017) doi: 10.1007/s1080101607306 (arxiv: 1601.08115).
 I. Cardinali, L. Giuzzi, "Enumerative Coding for Line Polar Grassmannians with Applications to Codes", Finite Fields Appl. 46, 107138 (2017) doi: 10.1016/j.ffa.2017.03.005 (arxiv: 1412.5466).
 I. Cardinali, L. Giuzzi, "Geometries arising from trilinear forms on lowdimensional vector spaces", Adv. Geom. to appear (arxiv: 1703.06821).
Preprints
 J.W.P. Hirschfeld, "Algebraic Geometry over a Field of Positive Characteristic  Appunti curati dal dott. L. Giuzzi", Quaderno del Seminario Matematico di Brescia n.16/98.
 L. Giuzzi, "Size of Hermitian intersections", Quaderno del Seminario Matematico di Brescia n.07/2001.
 L. Giuzzi, "Intersections of Hermitian surfaces/1: configurations", Quaderno del Seminario Matematico di Brescia n.11/2001.
 L. Giuzzi, "Intersections of Hermitian surfaces/2: matrices", Quaderno del Seminario Matematico di Brescia n.14/2001.
 L. Giuzzi, "Looking for ovoids of the Hermitian surface: a computational approach", Quaderno del Seminario Matematico di Brescia n.33/2002, (arxiv: 1210.2600).
 L. Giuzzi, "A family of ovoids of the Hermitian Surface in $PG(3,q^2)$ with $q\geq 5$ odd", Quaderno del Seminario Matematico di Brescia n.09/2003.
 L. Giuzzi, A. Sonnino, "Alcune note introduttive sulla crittografia", Quaderno del Seminario Matematico di Brescia n.01/2006.
 I. Cardinali, L. Giuzzi, "Minimum distance of Line Orthogonal Grassmann Codes in even characteristic", (arxiv: 1605.09333).
 I. Cardinali, L. Giuzzi, A. Pasini, "On transparent embeddings of pointline geometries", (arxiv: 1611.07877).
 I. Cardinali, L. Giuzzi, " Line Hermitian Grassmann Codes and their Parameters", (arxiv: 1706.10255).
Books
 L. Giuzzi, "Hermitian varieties over finite fields", DPhil thesis under the supervision of Prof. J.W.P. Hirschfeld (University of Sussex).
 L. Giuzzi, "Codici correttori", UNITEXT Springer Verlag 27 (2006), ISBN: 88–470–0539–6.
Proceedings/Extended Abstracts
 I. Cardinali, L. Giuzzi, "Some results on caps and codes related to orthogonal Grassmannians – a preview", Electron. Notes Discrete Math. 40, 139–144 (2013) ISSN: 15710653, \doi{10.1016\/\j.endm.2013.05.026}.
 I. Cardinali, L. Giuzzi, "Polar Grassmannians and their Codes", Extended Abstract accepted for the MEGA2015 conference (2015),(arxiv: 1509.07686).
Other printed works
 L. Giuzzi, "Gruppi di Frobenius e strutture geometriche associate" (Frobenius groups and associated geometrical structures, in Italian): tesi di laurea at Università Cattolica di Brescia, under the supervision of Prof. S. Pianta.
Patents
 L. Giuzzi, G. Korchmàros, A. Sonnino, "Perfezionamenti nella crittografia a chiave pubblica basata su curve ellittiche", patent no. 0001379714, August 2010.
Selected schools and workshops
 July 1995: "Summer school on combinatorics and finite geometry", Potenza.
 July 1996 "Summer school Giuseppe Tallini on finite geometry", Brescia.
 August 1996, "Scuola di Matematica Interuniversitaria", Perugia.
 September 1997, "Summer school on finite geometry", Potenza.
 April 1998, Intensive course "Galois Geometry and Generalised Polygons", Ghent.
 July 1998, "Advanced school on Combinatorial Geometry", Cortona.
 August 1998, NATO Advanced Study Institute "Difference sets, sequences and their correlation properties", Bad Winsheim.
 September 1998, "Summer school Giuseppe Tallini on finite geometry", Brescia.
 November 1998, Workshop "SUNCAGe'98", Caserta.
 May 1999, summer school "Methods of discrete mathematics: Association schemes, Lattices and Codes", Braunschweig.
 June 1999, Socrates Intensive programme "Finite geometries and their automorphisms", Potenza.
 September 1999, "International meeting on coding theory and cryptography", Medina del Campo (ES).
 October 2000, "ECC 2000: 4th workshop on elliptic curve cryptography", Essen.
 May 2001, "CHES 2001: Workshop on cryptographic hardware and embedded solutions", Paris.
 July 2002, "Summer school Giuseppe Tallini" on finite geometry, Brescia.
 September 2005, "Summer school Giuseppe Tallini on Combinatorial Geometries", Potenza.
 10–16 September 2006, "Second Irsee Conference on Finite Geometries", Irsee (DE).
 24–29 September 2007, "XVIII Congresso UMI", Bari (IT).
 22–28 June 2008, "Combinatorics 2008", Brescia (IT).
 13–17 June 2009, "Fq9 – Finite Fields and their Applications", Shannon Institute, Dublin (IE).
 30 May – 4 June 2010, "Fourth Pythagorean Conference", Corfu (GR).
 5–11 June 2010, NATO Advanced Study Institute, "Information Security and Related Combinatorics", Opatija (HR).
 27 June – 3 July 2010, "Combinatorics 2010", Verbania (IT).
 19–25 June 2011, "Third Irsee Conference on Finite Geometries", Irsee (DE).
 10–16 July 2011, "Fq10 – Finite Fields and their Applications", Ghent (BE).
 6–10 February 2012, "Incidenge geometry and buildings", Ghent (BE).
 13–14 February 2012, "Giornate di Geometria", Vicenza (IT).
 9–15 September 2012, "Combinatorics 2012", Perugia (IT).
Conference talks
 6 November 1997, "Spazi co–Minkowski e loro struttura di riflessione", GNSAGA national meeting, Perugia (IT).
 19 June 1998, "CoMinkowski spaces and their reflection structure", Combinatorics '98, Palermo (IT).
 31 May 2000, "Intersection numbers for Hermitian varieties", Combinatorics 2000, Gaeta (IT).
 19 July 2000, "Groups stabilising the intersection of two classical unitals", Fourth Isle of Thorns conference of finite geometries, Isle of Thorns (UK).
 5 July 2001, "A short characterisation of classical unitals", 18th British Combinatorial Conference, Brighton.
 8 June 2002, "Ovoids of the Hermitian surface", Combinatorics 2002, Maratea (IT).
 3 October 2003, "A family of ovoids of the Hermitian Surface in $PG(3,q^2)$ with odd $q\geq 5$", International Symposium on Graphs, Designs and Applications 2003, Messina (IT).
 26 June 2006, "LDPC Codes from Projective Spaces", Combinatorics 2006, Ischia (IT).
 30 January 2010, "Geometry and codes", workshop in honour of Prof. M. Marchi, on occasion of his 70th birthday.
 29 June 2010, "On samplings of graphs", Combinatorics 2010, Verbania (IT).
 12 July 2011, "Unitals in $PG(2,q^2)$ with a large $2$point stabiliser", Fq10 — 10th International Conference of Finite Fields and their Applications, Ghent (BE).
 10 September 2012, "Caps and codes from Polar Grassmannians", Combinatorics 2012, Perugia (IT).
 19 September 2013, "Linear codes from orthogonal Grassmannians", Conference on Random network codes and Designs, Ghent (BE).
 5 June 2014, "On Line Polar Grassmann Codes", Combinatorics 2014, Gaeta (IT).
 19 June 2014, "Linear Polar Grassmann Codes", Algebra, codes and Networks, Bordeaux (FR).
 18 September 2014, "Intersection of Hermitian Surfaces and quadrics", Finite Geometries — Fourth IRSEE conference, Kloster Irsee (DE).
 16 March 2015, "Polar Grassmann codes — Part II", ALCOMA15, Kloster Banz (DE).
 19 June 2015, "Polar Grassmannians and their Codes", MEGA2015, Trento (IT).
 3 June 2016, "Implementing Polar Gra{\ss}mann Codes", Combinatorics 2016, Maratea (IT).
 June 2017, "Hermitian Line Polar Gra{\ss}mann Codes", Fq13, Gaeta (IT).
 September 2017, "Transparent embeddings of pointline geometries", Fifth Irsee Conference, Irsee (DE).
Seminars
All the seminars in this section have been given upon invitation by the hosting institutions. June 1997, "Kinematic spaces derived from Frobenius Groups", School of Mathematical Sciences, University of Sussex.
 23 and 24 September 1998, "Cryptography and finite geometries (I and II)", Mathematics department at Università Cattolica del Sacro Cuore, Brescia.
 16 October 1998, "Publickey cryptosystems and elliptic curves", School of Mathematical Sciences, University of Sussex, Brighton.
 24 November 1999, "Of entropy and keys", School of Mathematical Sciences, University of Sussex, Brighton.
 10 February 2000, public seminar in "Trust, information and privacy: theory of secret writings and their legal issues", Università Cattolica del Sacro Cuore, Brescia.
 23 May 2001, "Intersection of Hermitian varieties", Università di Brescia.
 5 December 2001, "Hermitian varieties over finite fields: why?", Università Cattolica del Sacro Cuore, Brescia.
 10 April 2002, "Caps and ovoids in the Hermitian surface: a comutational approach", Università di Brescia.
 11 July 2002, "An introduction to modern cryptography: from RSA to AES", Summer School "Giuseppe Tallini" 2002, Brescia.
 5 February 2003, "Introduction to Coding theory", Università Cattolica del Sacro Cuore, Brescia.
 12 February 2003, "From Codes to Designs", Università Cattolica del Sacro Cuore, Brescia.
 6 November 2003, "Incidence configurations in the Hermitian surface", Giornate di Geometria, Università Cattolica del Sacro Cuore, Brescia.
 20 November 2003, "Mathematics plus secrecy", Museum of Sciences, Brescia.
 4 May 2004, "Finite geometry and Cryptography", Politecnico di Bari.
 6 May 2004, "Finite fields in C++", Dipartimento di Matematica, Università di Napoli.
 9 September 2005, "A defense of cryptography", Università della Basilicata.
 29 May 2006 and 31 May 2006, "An introduction to algebraic decoding of cyclic codes", Facoltà di Ingegneria, Università di Brescia.
 25 October 2006, "Security, privacy and secrecy: several faces of cryptography", Dipartimento di matematica, Università di Bari.
 21 November 2006, "Reed–Solomon codes", Facoltà di Ingegneria, Università di Brescia.
 23 Novembre 2006, "Cryptography of (in)security", Facoltà di Ingegneria, Università di Brescia.
 4 and 6 December 2006, "Uses of cryptography: from secrecy to security", Facoltà di Economia di Bari.
 5 December 2008, "Elliptic Curves: theory", Università Cattolica, Brescia
 13 February 2009, "Elliptic Curves: practice", Università Cattolica, Brescia.
 8 April 2009, "Non–binary BCH codes", Dipartimento di Matematica, Università Federico II, Napoli.
 30 January 2010, "Geometry and codes", Worshop in honor of M.Marchi, Università Cattolica, Brescia.
 29 October 2010, "Mathematics and secrets", Università Cattolica, Brescia.
 From 15 till 21 May 2011 L. Giuzzi has been a guest of the Mathematics Department of the University of Gent (BE).
 15 March 2012, "Non–linear codes", Dipartimento di Matematica, Università Federico II, Napoli.
 24 April 2012, "Algebraic and geometric methods in cryptography", Dipartimento di Matematica, Politecnico di Milano.
 21 January 2014, "Strutture geometriche, combinatorica ed applicazioni", DICATAM, Università di Brescia.
Summary of research interests
 Incidence geometries.
 Coding theory and Cryptography.
 Finite geometries and varieties over finite fields.
 Graph designs.
Computer skills
 Very good knowledge of Linux (since 1994);
 Experience as system administrator of Solaris (from 2.5 up to 2.8) and Digital Unix (4.0A  4.0E) systems;
 Good knowledge of C and C++;
 Good knowledge of HTML, XHTML and CSS;
 Experience with perl and PHP;
 Working knowledge of several Computer Algebra systems, amongst which there are GAP, Maxima, Axiom, Singular and pari/GP.
Abstracts of the main papers
Hermitian varieties over finite fields
L. Giuzzi, University of Sussex (2000)
A projective space ${{PG}}(n,q)$ admits at most three types of polarity: orthogonal, symplectic and unitary. The absolute points of an orthogonal polarity constitute a nondegenerate quadric in ${{PG}}(n,q)$; for a symplectic polarity, all the points of ${{PG}}(n,2^t)$ are absolute; the locus of all absolute points of a unitary polarity is a nondegenerate Hermitian variety.Nondegenerate Hermitian varieties are unique in ${{PG}}(n,q)$ up to projectivities. However, two distinct Hermitian varieties might intersect in many different configurations. Our aim in this thesis is to study such configurations in some detail.
In Chapter 1 we introduce some background material on finite fields, projective spaces, collineation groups and Hermitian varieties.
Chapter 2 deals with the twodimensional case. Kestenband has proven that two Hermitian curves may meet in any of seven pointline configurations. In Section 2.1, we present this classification. In Section 2.2, we verify that any two configurations belonging to the same class are, in fact, projectively equivalent and we determine the linear collineation group stabilizing each of them. Such a group is usually quite large and is transitive on almost all the points of the intersection. A subset $U$ of ${{PG}}(2,q)$ such that any line of the plane meets $U$ in either $1$ or $\sqrt{q}+1$ points is called a unital. A Hermitian curve is a classical unital. However, there exist nonclassical unitals as well. In Section 2.3 we present a short proof of a characterisation of the Hermitian curve as the unital stabilized by a Singer subgroup of order $q\sqrt{q}+1$.
In Chapter 3, we describe the pointlineplane configurations arising in dimension $3$ from intersecting two Hermitian surfaces. Our approach consists first in determining some combinatorial properties the configurations have to satisfy and then in actually constructing all the possible cases. Section 3.1 presents the list of all possible intersection classes; after some more technical results in Section 3.2, in Section 3.3 we construct linear systems of Hermitian surfaces yielding the wanted configurations for any class. In this chapter we deal with intersections which contain at least $\sqrt{q}+1$ points on a line.
Chapter 4 is divided into two independent sections: in Section 4.1 we study the determinantal variety of all the $(n+1)\times(n+1)$ Hermitian matrices as a hypersurface of ${{PG}}(n^2+2n,\sqrt{q})$. From the study of such a variety we are able to determine the list of all possible intersection sizes for any dimension $n$. In Section 4.2 we present some computer code in order to produce pencils of Hermitian varieties in ${{PG}}(n,q)$. This code, however, is able to provide useful results only for small values of $n$ and $q$. Some possible improvements, both from the theoretic standpoint and the computational one, are suggested.
Codici Correttori
L. Giuzzi, UNITEXT Springer Verlag 27 (2006)
This book (in italian) is a selfcontained introduction to coding theory for undergraduate students in engineering, mathematics or computer science. The main topics are The theory of linear block codes, with special regards to algebraic constructions and algebraic decoding;
 Construction of codes from designs and links between projective geometry and coding theory;
 Asymptotic bounds;
 An introduction to AlgebraicGeometry codes;
 LDPC codes and Tanner graphs;
 Convolutory codes.
Collineation Groups of the Intersection of Two Classical Unitals
L. Giuzzi, J. Comb. Des. 9: 445–459 (2001).
Kestenband proved that there are only seven pairwise nonisomorphic Hermitian intersections in the Desarguesian projective plane $\mathrm{PG}(2,q^2)$ of square order $q^2$. His classification is based on the study of the minimal polynomials of the matrices associated with the curves and leads to results of purely combinatorial nature: in fact, two Hermitian intersections from the same class might not be projectively equivalent in $\mathrm{PG}(2,q^2)$ and might have different collineation groups. The projective classification of Hermitian intersections in $\mathrm{PG}(2,q^2)$ is the main goal in this paper. It turns out that each of Kestenband's classes consists of projectively equivalent Hermitian intersections. A complete classification of the linear collineation groups preserving a Hermitian intersection is also given.CoMinkowski Spaces, their Reflection Structure and Kloops
L.Giuzzi, H. Karzel, Discrete Math. 255: 161–179 (2002).
In this work an infinite family of Kloops is constructed from the reflection structure of coMinkowski planes and their properties are analysed.A Characterisation of Classical Unitals
L. Giuzzi, J. Geom., 74: 86–89 (2002).
A short proof is given of the following result: A unital in $\mathrm{PG}(2,q)$ is classical if and only if it is preserved by a cyclic linear collineation group of order $q\sqrt{q}+1$.}Ovoids of the Hermitian Surface in Odd Characteristic
L. Giuzzi, G. Korchmáros, Adv. Geom., Special Issue (2003), S49–S58.
We construct a new ovoid of the polar space arising from the Hermitian surface of $\mathrm{PG}(3,q^2)$ with $q\geq 5$ odd. The automorphism group $\Gamma$ of such an ovoid has a normal cyclic subgroup $\Phi$ of order $\frac{1}{2}(q+1)$ such that orbits on the ovoid, one of size $q+1$ and two of sizeOn the Intersection of Hermitian Surfaces
L. Giuzzi, J. Geom., 85: 49–60 (2006).
We provide a description of the configuration arising from intersection of two Hermitian surfaces in $\mathrm{PG}(3,q)$, provided that the linear system they generate contains at least a degenerate variety.A Geometric Construction for Some Ovoids of the Hermitian Surface
L. Giuzzi, Results Math. 49: 81–88 (2006).
Multiple derivation of the classical ovoid of the Hermitian surface is a well known, powerful method for constructing large families of non classical ovoids of ${\mathcal H}(3,q^2)$. In this paper, multiple derivation is generalised and applied to nonclassical ovoids. A resulting new family of ovoids is investigated.Orthogonal Arrays from Hermitian Varieties
A. Aguglia, L. Giuzzi, Innov. Incidence Geom. 5: 129–144 (2007).
A simple orthogonal array $\mathrm{OA}(q^{2n1},q^{2n2}, q,2)$ is constructed by using the action of a large subgroup ofConstruction of a $3$–dimensional MDS Code
A. Aguglia, L. Giuzzi, Contrib. Discrete Math. 3 (1), 39–46 (2007).
In this paper, we describe a procedure for constructing $q$ary $[N,3,N2]$MDS codes, of length $N\leq q+1$ (for $q$ odd) or $N\leq q+2$ (for $q$ even), using a set of nondegenerate Hermitian forms in $\mathrm{PG}(2,q^2)$.An Algorithm for Constructing some Maximal Arcs in ${{PG}}(2,q^2)$
A. Aguglia, L. Giuzzi, Results Math. 52 no. 1–2: 17–33 (2008).
In 1974, J. Thas constructed a new class of maximal arcs for the Desarguesian plane of order $q^2$. The construction relied upon the existence of a regular spread of tangent lines to an ovoid in ovoid. In this paper, we describe an algorithm for obtaining a possible representation of such arcs in $\mathrm{PG}(2,q^2)$.Algebraic Curves and Maximal Arcs
A. Aguglia, L. Giuzzi, G. Korchmáros, J. Algebraic Combin. 28: 531–544 (2008)/
A lower bound on the minimum degree of the plane algebraic curves containing every point in a large pointset $\mathcal K$ of the Desarguesian plane $\mathrm{PG}(2,q)$ is obtained. The case where $\mathcal K$ is a maximal $(k,n)$arc is considered to greater extent.On the Nonexistence of Certain Hyperovals in Dual André Planes of Order $2^{2k}$
A. Aguglia, L. Giuzzi, Electron. J. Combin. 15(1): N37 (2008).
No oval contained in a regular hyperoval of the Desarguesian plane $\mathrm{PG}(2,q^2)$, $q$ even, is inherited by a Moulton plane of order $q^2$.LDPC Codes from Singer Cycles
L. Giuzzi, A. Sonnino, Discrete Appl. Math. 157: 1723–1728 (2009).
The main goal of coding theory is to devise efficient systems to exploit the full capacity of a communication channel, thus achieving an arbitrarily small error probability. Low Density Parity Check (LDPC) codes are a family of block codes – characterised by admitting a sparse parity check matrix – with good correction capabilities. In the present paper the orbits of subspaces of a finite projective space under the action of a Singer cycle are investigated. The incidence matrix associated to each of these structures yields an LDPC code in a natural manner.Construction of Unitals in Desarguesian Planes
A. Aguglia, L. Giuzzi, G. Korchmáros, Discrete Math. 310 (22): 3162–3167 (2010).
We present a new construction of nonclassical unitals from a classical unital $\mathcal U$ in $\mathrm{PG}(2,q^2)$. The resulting nonclassical unitals are B–M unitals. The idea is to find a nonstandard model $\Pi$ of $\mathrm{PG}(2,q^2)$ with the following three properties: points of $\Pi$ are those of $\mathrm{PG}(2,q^2)$;
 lines of $\Pi$ are certain lines and conics of $\mathrm{PG}(2,q^2)$;
 the points in $\mathcal U$ form a nonclassical BM unital in $\Pi$.