Professor Ray Hill

Honorary Professor of Mathematics

  • Newton Building Room LG10
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Office Times

Friday 11.00am-1.30pm


I obtained my BSc (First Class Honours) in Mathematics from the University of Manchester in 1967. I gained an MSc (1968) and PhD (1971) in Mathematics from the University of Warwick, under the supervision of Professor Sandy Green. After working for two years as a Temporary Lecturer at the University of Nottingham, I was appointed Lecturer in Mathematics at the University of Salford in 1973. I was promoted to Senior Lecturer in 1985, to Reader in 1992, and to a Personal Chair in 1999. I took early retirement in 2000 and now have an appointment at Salford as Honorary Professor.

I have published over 50 research papers on group theory, finite geometry, coding theory and combinatorics. My book “A First Course in Coding Theory” (Oxford University Press, 1986) has sold over 12,000 copies. Since 2002, this book has been the core text for the Open University Master’s Module “M836 – Coding Theory”.

I have given over 60 conference talks, including main invited speaker at the British Association for the Advancement of Science (1980 and 1989), Fortieth British Mathematical Colloquium (1988), Second IMA Conference on Cryptography and Coding (1989), Fifteenth British Combinatorial Conference (1995), and many overseas conferences. I have given invited seminars at over 30 universities, including Oxford (three times), Cambridge, and Massachusetts Institute of Technology.

My external appointments include Membership (1983-87) and Secretary (1984-87) of the British Combinatorial Committee, Editorial Board of “Serdica” (1994-2004), Editorial Board of “Designs, Codes and Cryptography” (2003-present), Member of Organizing Committee for a series of International Workshops on Optimal Codes held in Bulgaria (1995, 1998, 2001), Member of Organizing Committees for IMA Conferences “Applications of Finite Fields” and “Applications of Combinatorial Mathematics” (1994), and EPSRC Panel Member (1998).

I supervised PhD students John Mason, David Newton, Karen Traynor, Phil Greenough, Pawel Lizak, Chris Jones, Jehangir Karim and Chris Love.

I have had the pleasure of collaborating with many distinguished mathematicians, such as:

  • Ivan Landjev (Bulgarian Academy of Sciences), who spent two years with me at Salford , first as a Royal Society Visiting Fellow, then as an EPSRC funded Visiting Fellow, working on optimal code problems.
  • Emil Kolev (Bulgarian Academy of Sciences), who was also a year-long Royal Society Visiting Fellow.
  • Elwyn Berlekamp (MIT), with whom Jehangir Karim and I solved a famous long-standing problem of Ulam (how to win a game of “Twenty Questions” against a liar).
  • Thann Ward (University of Virginia), who has worked with me at Salford on a number of short visits. Together with Ivan Landjev and Simeon Ball, Thann and I have carried out some novel research on multi-arcs in finite geometries and their links with codes.
  • Tatsuya Maruta (University of Osaka), who spent a year (and several shorter visits) with me at Salford, and who has greatly extended my initial work on extendibility of codes.
  • Leo Storme (University of Ghent), with whom, together with other collaborators, we have made progress on cap problems in finite geometry.

My main research career has centred on coding theory and finite geometry, and particularly on the interplay of these two topics.

Since my early retirement in 2000, I have branched into a quite new direction, namely the use (or misuse) of probability and statistics in criminal trials. I worked for the defence teams of mothers such as Sally Clark, Angela Cannings and Trupti Patel, who not only suffered the tragic sudden unexplained deaths of two or more babies, but were then, on the basis of flawed statistical evidence, accused of their murder. My work towards proving the innocence of these mothers has led to publications in medical and statistical journals, as well as invitations to speak at mathematical, statistical, medical and legal conferences and seminars. The work has attracted a fair amount of media coverage and has also been commended by Sir Herman Bondi in a letter to “Nature”, by Barry Lewis in his presidential address as President of the Mathematical Association, by Angela Saini in an article “Justice you can count on” in New Scientist (24 October 2009), and by Sir Mervyn King in his Inaugural Lecture (on probability) as Governor of the Bank of England.  My papers (listed below) on this topic are available in electronic form on request from

  • “Cot death or murder – weighing the probabilities”, Developmental Physiology Conference, June 2002.
  • “Multiple sudden infant deaths – coincidence or beyond coincidence?”, Paediatric and Perinatal Epidemiology, 18 (2004), 320-326.
  • “Reflections on the cot death cases”, Significance, 2 (2005), 13-15.


I am currently supervising a final year project on Cryptography.

Research Interests

Coding theory, Finite geometry, Combinatorics, Group theory, Probability and Statistics.

Qualifications and Memberships

BSc in Mathematics, University of Manchester, 1967.

MSc in Mathematics, University of Warwick, 1968.

PhD in Mathematics, University of Warwick, 1971.

Member of London Mathematical Society.

Member of Institute for Combinatorics and its Applications.


Book:  A First Course in Coding Theory, Oxford University Press (1986).

Journal and other publications: 

JA Green and R Hill. On a theorem of Fong and Gaschutz, J London Math Soc (2)1 (1969) 573-576.

R Hill.  Rank 3 permutation groups with a regular normal subgroup, PhD thesis, University of Warwick (1971).

R Hill.  The largest size of cap in S5,3, Rend Acc Naz Lincei (8) 54 (1973), 378-384.

R Hill.  Caps and groups, Atti del Colloquio Internazionale sulle Teorie Combinatorie, Rome 1973, Vol II (1976), 389-394.

R Hill.  Caps and codes, Discrete Mathematics 22 (1978), 111-137.

R Hill.  Packing problems in Galois geometries over GF(3), Geometriae Dedicata 7 (1978), 363-373.

R Hill.  Some results concerning linear codes and (k,3)-caps in three dimensional Galois space, Math Proc Camb Phil soc 94 (1978), 191-205.

R Hill and JRM Mason.  On (k,n)-arcs and the falsity of the Lunelli-Sce conjecture, Proc of Second Isle of Thorns Conference on Finite Geometries and Designs 1980, edited by PJ Cameron, JWP Hirschfeld and DR Hughes, LMS Lecture Note Series 49, Cambridge University Press (1981), 153-168.

R Hill and RW Irving.  On group partitions associated with lower bounds for symmetric Ramsey numbers, European Journal of Combinatorics 3 (1982), 35-50.

R Hill.  On Pellegrino’s 20-caps in S4,3, Annals of Discrete Mathematics 18 (1983), 433-448.

R Hill.  Solution of Problem 66.H, Mathematical Gazette 67, No 440 (1983), 141.

R Hill.  Some problems concerning (k,n)-arcs in finite projective planes, Rendiconti del Seminario Matematico di Brescia 7 (1984), 367-383.

R Hill.  Solution of Problem 85-7, The Mathematical Intelligencer, Vol 9, No 1 (1987), 63-64.

R Hill.  Ramanujan – his life and work, Mathematical Spectrum 20 (1987), 1-8.

R Hill and DE Newton.  Some optimal ternary linear codes, Ars Combinatoria 25A (1988), 61-72.

R Hill.  Counting non-overlapping pairs, Mathematical Gazette 73, No 464 (1989), 120-122.

R Hill and KL Traynor.  The non-existence of certain binary linear codes, IEEE Trans on Information Theory 36 (1990), 917-922.

R Hill.  Error-correcting codes, Part I, Mathematical Spectrum 22 (1989/90), 94-102.

R Hill.  Error-correcting codes, Part II, Mathematical Spectrum 23 (1990/91), 14-22.

R Hill.  Cricket competitions and random derangements, Bulletin of the Institute of Combinatorics and its Applications, 2 (1991), 82-85.

R Hill.  Counting and the reflection principle, Mathematical Gazette 75 (1991), 308-312.

R Hill.  Optimal linear codes, in Cryptography and Coding II, edited by C Mitchell, Oxford University Press (1992), 75-104.

PP Greenough and R Hill.  Optimal ternary quasi-cyclic codes, Designs, Codes and Cryptography 2 (1992), 81-91.

R Hill and DE Newton.  Optimal ternary linear codes, Designs, Codes and Cryptography 2 (1992), 137-157.

R Hill and PP Greenough.  Optimal quasi-twisted codes, in Proc International Workshop on Algebraic and Combinatorial Coding Theory, Voneshta Voda (1992), 92-97.

R Hill and JP Karim.  Searching with lies: the Ulam problem, Discrete Mathematics 106/107 (1992), 273-283.

PP Greenough and R Hill.  Optimal linear codes over GF(4), Discrete Mathematics 125 (1994), 187-199.

M van Eupen and R Hill.  An optimal ternary [69,5,45] code and related codes, Designs, Codes and Cryptography 4 (1994), 271-282.

I Landgev, R Hill and P Lizak.  Optimal quaternary codes of dimension 4 and 5, Proc of Fourth International Workshop on Algebraic and Combinatorial Coding Theory, Novgorod, Russia (1994), 98-101.

R Hill and P Lizak.  The non-existence of [32,5,22] codes over GF(4), Proc of International Workshop on Optimal Codes, Sozopol, Bulgaria (1995), 90-95. 

R Hill.  Searching with lies, in Surveys in Combinatorics, edited by P Rowlinson, Cambridge University Press (1995), 41-70.

R Hill and P Lizak.  Extensions of linear codes, Proc of IEEE International Symposium on Information Theory, Whistler, Canada (1995), 345.

I Landgev, T Maruta and R Hill.  On the nonexistence of quaternary [51,4,37] codes, Finite Fields and their Applications 2 (1996), 96-110.

R Hill and I Landgev.  On the nonexistence of some quaternary codes, in Applications of Finite Fields, edited by D Gollmann, Oxford University Press (1996), 85-98.

SM Elzaidi and R Hill.  Cubes and inverses of magic squares, Math Gazette 489 (1996), 565-567.

R Hill and P Lizak.  Some geometric constructions of optimal quaternary codes, Proc of IEEE International Symposium on Information Theory, Ulm, Germany (1997), 114.

R Hill.  Giusseppe Tallini – a survey of some of his papers and personal memories, Results in Mathematics 32 (1997), 254-259.

R Hill.  The pigeonhole principle, letter to OR Newsletter, March 1998, p 17.

R Hill and CM Jones.  The nonexistence of ternary [47,6,29] codes, Proc of 2nd International Workshop on Optimal Codes, Sozopol, Bulgaria (1998), 90-96.

R Hill, JP Karim and ER Berlekamp.  The solution of a problem of Ulam on searching with lies, Proc of IEEE International Symposium on Information Theory, MIT Cambridge, USA (1998), 244.

R Hill and E Kolev.  A survey of recent results on optimal linear codes, in Combinatorial Designs and their Applications, CRC Press Research Notes in Mathematics (1999), 127-152.

E Kolev and R Hill.  An improved lower bound on the covering number K2(9,1), Discrete Mathematics 197/198 (1999), 483-489.

R Hill.  An extension theorem for linear codes, Designs, Codes and Cryptography 17 (1999), 151-157.

R Hill.  Book review of “Projective Geometries over Finite Fields” by JWP Hirschfeld, Bull London Math Soc 31 (1999), 759-760.

R Hill, I Landjev, C Jones, L Storme and J Barat.   On complete caps in the projective geometries over F3,  Journal of Geometry 67 (2000), 127-144.

J Barat, Y Edel, R Hill, C Jones, I Landjev and L Storme, Caps in PG(5,3) and PG(6,3),  Proc of 7th International Workshop on Algebraic and Combinatorial Coding Theory, Bansco, Bulgaria (2000), 65-67.

M Grannell, T Griggs and R Hill.  The triangle chromatic index of Steiner triple systems, Australasian Journal of Combinatorics 23 (2001), 217-230.

R Hill and C Love.  On the (22,4)-arcs in PG(2,7) and related codes,  Proc of Third Euro Workshop on Optimal Codes and Related Topics, Sunny Beach, Bulgaria (2001), 96-101.

R Hill and T Maruta.  The nonexistence of [206,4,164]5 codes, Proc of Third Euro Workshop on Optimal Codes and Related Topics, Sunny Beach, Bulgaria (2001), 111-116.

S Ball, R Hill, I Landjev and H Ward.  On (q2+q+2, q+2)-arcs in the projective plane PG(2,q), Designs, Codes and Cryptography 24 (2001), 205-224.

I Landjev, A Rousseva, T Maruta and R Hill.  On optimal codes over the field with five elements, Designs, Codes and Cryptography 29 (2003), 165-175.

R Hill and C Love.  On the (22,4)-arcs in PG(2,7) and related codes, Discrete Mathematics 266 (2003), 253-261.

J Barat, Y Edel, R Hill and L Storme.  On complete caps in the projective geometries over F3, II: new improvements, Journal of Combinatorial Mathematics and Combinatorial Computing 49 (2004), 9-31.

R Hill.  Multiple sudden infant deaths – coincidence or beyond coincidence? Paediatric and Perinatal Epidemiology 18 (2004), 320-326.

R Hill.  Reflections on the cot death cases,  Significance 2 (2005), 13-15.

[This paper also appeared, with additional comment by the editor, as a chapter in the book “Statistics in the Law”, edited by JB Kadane, Oxford University Press (2008), 399-411.]

R Hill and H Ward.  A geometric approach to classifying Griesmer codes, Designs, Codes and Cryptography 44 (2007), 169-196.