Dr. Sophie Huczynska

Current Position

I am a Senior Lecturer in Pure Mathematics at the School of Mathematics and Statistics at the University of St Andrews. Until October 2010, I was a Royal Society Dorothy Hodgkin Research Fellow . My research is mainly in the areas of combinatorics and finite fields; I also have interests in algebra and theoretical computer science. I am a member of the Centre for Interdiscipinary Research in Computational Algebra (CIRCA).

Please note: I work on a part-time basis. I can be contacted in the department on Mondays and Wednesdays, and until mid-afternoon on all other days.

Teaching and Departmental Service

Research interests

I am interested in a range of areas in, and related to, combinatorics. My original background is in the structural theory of finite fields, and I have worked on various questions related to the existence of primitive normal bases for finite fields. I am also interested in applications of finite fields to combinatorial designs, and to coding theory and cryptography.

Recently I have been investigating connections between combinatorics and cryptography, via the concept of external difference families and their generalizations. These have a natural connection to optimal AMD codes. Many of the constructions in this area use cyclotomic methods from finite fields, but there are rich connections with other areas of combinatorics, finite geometry and group theory. I was recently funded by a Carnegie Research Incentive Grant to work on this with my collaborators.

Another strand of my research is viewing combinatorial objects as relational structures, and asking questions about partial orders on these structures. In particular, I have been investigating the homomorphic image order, a natural analogue of the substructure order, with a focus on the setting of graphs and digraphs.

Recent/ forthcoming events

  • Member of the Scientific Comittee for Fq14 - 14th International Conference on Finite Fields and their Applications (Vancouver, June 2019).
  • Member of the organizing committees of various conferences, including the 70th British Mathematical Colloquium (St Andrews, 2018), and Fq12 - the 12th International Conference on Finite Fields and their Applications (Saratoga Springs NY, 2015).
  • Co-host (with Nik Ruskuc) of the 3rd Scottish Combinatorics Meeting at St Andrews in 2017.
  • Invited speaker at various conferences and meetings: eg London Combinatorics Colloquia, Fq11 - the 11th International Conference on Finite Fields and their Applications.

    Academic service

    Public understanding


  • Fiona Brunk:(2009) completed her PhD on "Intersection problems in combinatorics", jointly supervised by Nik Ruskuc and myself.
  • I was second supervisor for Anna Schroeder, who completed her PhD under the primary supervision of Colva Roney-Dougal.
  • I am currently second supervisor for Nseobong Uto, whose first supervisor is Rosemary Bailey.



  • "Surveys in Combinatorics", 2009, S.Huczynska, J.D.Mitchell and C.Roney-Dougal (eds). LMS Lecture Note Series, 365, Cambridge University Press 2009.

    Book chapters

  • "Existence results for finite field polynomials with specified properties", in Finite Fields and Their Applications: Character Sums and Polynomials, P. Charpin, A. Pott and A. Winterhof (eds), De Gruyter, Berlin, Boston, 2013.

    Journal papers

  • "Strong external difference families in abelian and non-abelian groups ", S.Huczynska, C. Jefferson and S. Nepsinska, preprint (available on ArXiv).
  • "Characterising bimodal collections of sets in finite groups", S.Huczynska and M.B. Paterson, Archiv der Mathematik, in press.
  • "Weighted external difference families and R-optimal AMD codes", S.Huczynska and M.B. Paterson, Discrete Mathematics 342, 3 (2019), p.855-867.
  • "Existence and non-existence results for strong external difference families", S.Huczynska and M.B. Paterson, Discrete Mathematics. 341, 1 (2018) p. 87-95
  • "Near-complete external difference families", J.A. Davis, S. Huczynska, and G.L. Mullen, Designs, Codes and Cryptography. 84, 3, p. 415-424 (2017) 10 p.
  • "On well quasi-order of graph classes under homomorphic image orderings", S.Huczynska and N. Ruskuc, European Journal of Combinatorics. 63, p. 164-175 (2017) 12 p.
  • "Well quasi-order in combinatorics: embeddings and homomorphisms'', S.Huczynska and N. Ruskuc, Surveys in combinatorics 2015, 261–293, London Math. Soc. Lecture Note Ser., 424, Cambridge Univ. Press, Cambridge, 2015.
  • "Homomorphic image orders on combinatorial structures", S.Huczynska and N.Ruskuc, Order 32 (2015), 205–226.
  • "Beyond sum-free sets in the natural numbers", S. Huczynska, The Electronic Journal of Combinatorics, Volume 21, Issue 1 (2014), P1.21
  • "Existence and properties of k-normal elements over finite fields", S. Huczynska, G.L. Mullen, D. Panario and D. Thomson, Finite Fields and Their Applications 124 (2013), 170--183.
  • "The strong primitive normal basis theorem", S.D. Cohen and S. Huczynska, Acta Arithmetica 143 (2010), 299--332.
  • "Some Erdo"s-Ko-Rado theorems for injections", F. Brunk and S. Huczynska, European Journal of Combinatorics 31 (2010), 839--860.
  • "Equidistant frequency permutation arrays and related constant composition codes", S. Huczynska, Designs, Codes and Cryptography 54 (2010), 109--120.
  • "The extent to which subsets are additively closed", S. Huczynska, G.L. Mullen and J. Yucas, Journal of Combinatorial Theory, Series A, 116 (2009), 831--843.
  • "Decomposing simple permutations, with enumerative consequences", R. Brignall, S. Huczynska and V. Vatter, Combinatorica 28 (2008), 385--400.
  • "Simple permutations and algebraic generating functions", R. Brignall, S. Huczynska and V. Vatter, Journal of Combinatorial Theory, Series A, 115, 423--441.
  • "Pattern classes of permutations via bijections between linearly ordered sets", S. Huczynska and N. Ruskuc, European Journal of Combinatorics 29 (2008), 118--139.
  • "Powerline communication and the 36 officers problem", S. Huczynska, Phil. Trans. R. Soc. A, 364 (2006).
  • "Frequency permutation arrays", S. Huczynska and G.L. Mullen, Journal of Combinatorial Designs 14 (2006), 463--478.
  • "Grid classes and the Fibonacci dichotomy for restricted permutations", S. Huczynska and V. Vatter, The Electronic Journal of Combinatorics 13 (2006)
  • "Primitive free cubics with specified norm and trace", S. Huczynska and S.D. Cohen, Transactions of the American Mathematical Society 355 (2003), 3099--3116.
  • "Primitive free quartics with specified norm and trace", S.D. Cohen and S. Huczynska, Acta Arithmetica 109 (2003), 359--385.
  • "The Primitive Normal Basis Theorem- without a computer", S.D. Cohen and S. Huczynska, Journal of the London Mathematical Society, 67 (2003), 41--56 .

    Conference papers

  • "Modelling equidistant frequency permutation arrays in constraints", Ian P. Gent, Paul McKay, Ian Miguel, Pete Nightingale and Sophie Huczynska, Proceedings of Principles and Practice of Constraint Programming(CP 2009).
  • "The Homer System", S. Colton and S. Huczynska, Automated Deduction � CADE-19: 19th International Conference on Automated Deduction, 2003, Franz Baeder (ed), Lecture Notes in Computer Science, Springer.

    Contact details:

    Sophie Huczynska
    School of Mathematics and Statistics
    University of St Andrews
    Mathematical Institute (Room 312)
    North Haugh
    St Andrews, Fife, KY16 9SS

    Tel: +44 (0)1334 463711
    E-mail: My departmental email address is: sh70 'at'st-andrews.ac.uk