Edward F. Assmus, Jr   (1931-1998)

Ed Assmus died suddenly at the age of 66, on the evening of Wednesday 18 March 1998 while participating in the meeting on Designs and Codes at Oberwolfach, Germany. Shocked and saddened, the other participants held a memorial service for him on the following day and brought the meeting to a swift close.

Ed retired from his position at Lehigh University in Bethlehem, Pennsylvania, in July 1994, after 28 years as a member of the faculty there. During that time he had become world-renowned for his work in codes and designs. When he started his research in codes, in 1962, most mathematicians were unacquainted with the subject, and his work did much to make coding theory recognized in the mathematical world.

Born on 19 April 1931 at Nutley, New Jersey, he was educated at Oberlin College and at Harvard University, where he obtained his doctorate in 1958. From 1958 to 1962 he was employed as Office of Naval Research Associate and Ritt Instructor at Columbia University. After that, and until 1966, he was a Lecturer at Wesleyan University. It was during that period that his interests shifted from homological algebra to applied combinatorics. He joined the faculty at Lehigh University in 1966 as associate professor and was promoted to professor in 1970. In May 1993 he was awarded the title of Distinguished Professor. He retired in July 1994. During his time at Lehigh he had many productive visiting positions in the U.S.A., Canada, and Europe. His excellence as a lecturer made him in demand as an invited speaker at many universities and conferences; he gave over 100 invited lectures on 80 separate occasions in mathematics departments and universities in the United States and in nine other countries. His publication list includes approximately 60 research papers and many book reviews. He had eleven doctoral students at Lehigh: E. P. Shaughnessy (1969), M. M. Guza (1973), H. E. Sachar (1973), D. P. Maher (1975), J. A. Mezzaroba (1975), C. J. Salwach (1976), B. R. Andriamanalimanana (1979), J. E. N. Sardi (1980), S. T. Dougherty (1992), K. J. Rose (1993), J. C. Janssen (1993). He was a reviewer for both Mathematical Reviews and Zentralblatt für Mathematik, and wrote occasional book reviews for periodicals. He was a member of the editorial board of the journals Designs, Codes and Cryptography, and the Electronic Journal of Combinatorics, and refereed for various mathematical journals, including Journal of Combinatorial Theory, Information and Control, Discrete Mathematics, American Mathematical Monthly, and IEEE Transactions on Information Theory. He was a member of the American Mathematical Society, the London Mathematical Society, the Society for Industrial and Applied Mathematics, the Mathematical Association of America, the American Association for the Advancement of Science, and the Institute of Combinatorics and its Applications.

After his retirement, he remained professionally active, traveling extensively and often invited to spend extended periods at various prestigious research establishments. In particular, he frequently spent some months at INRIA, near Versailles, working with Paul Camion and Pascale Charpin at Projet Codes.

Ed's doctoral thesis, under John Tate at Harvard in 1958, was on the homology of local rings. He proved that Tor^R(K,K), where R is a local ring and K the residue class field, is a Hopf algebra. Although much more is now known about these algebras, his results [[1]] are still cited in the homological algebra literature.

Within a few years his interests had changed, and he started to work on the then somewhat unfashionable area of algebraic coding theory. A long collaboration with H. F. Mattson, jr., resulted in a series of papers on the relationship between algebraic coding theory and classical combinatorial theory. Possibly the best known of the joint papers was [[16]], where they showed how combinatorial information can be extracted from codes. In particular, the central theorem of that paper shows how vectors of a fixed weight in a linear code can produce a t-design; this frequently quoted result is referred to as the ``Assmus-Mattson Theorem".

From 1962 to 1972, Ed and Mattson produced an annual report for the Air Force contract under which they worked. These reports, some of which are included in the list of publications, contained all their work for the year, and some parts were published in journals. The reports, which were widely circulated, and the expository paper [[23]] which explained some of their results, helped to interest mathematicians in coding theory. During the same period there were monthly meetings of a group centering around Andy Gleason, with other members Gene Prange, Vera Pless, John N. Pierce, Skip Mattson, and Ed. Usually guests also came, and frequent visitors were Elwyn Berlekamp, Jessie MacWilliams, and Neil Sloane. One of the highlights was Ed's presentation in spring 1970 of his results on the plane of order 10 to an appreciative audience of `members' and guests from Bell Labs.

Ed's interest in the relationship between codes and designs continued, and at a lecture in Oberwolfach in spring 1970, he pointed out the connection between the problem of the existence of a projective plane of order 10 and the existence of certain doubly-even, self-dual binary codes. His approach [[18],[21]], extended by several subsequent researchers, was the key to the proof by Lam¹ et al. (by computer, in 1989) that no such plane exists. Somewhat more generally, he established connections between symmetric designs of even order and their ovals on the one hand, and self-orthogonal codes on the other; this was joint work with J. H. van Lint [[32]].

A more recent collaboration with J. D. Key commenced in 1986 with the publication [[41]], in which a simple method for extending some Steiner 2-designs using ovals was described, producing a new class of Steiner 3-designs. This collaboration continued with further research into codes associated with designs, and the attempt to use coding theory as a classifying tool for the designs. This resulted in a series of papers [[45],[46],[47],[48],[52],[56]] on codes and planes, unitals, and Hadamard designs. The culmination of the joint work was a research monograph [[51]] on the link between designs and codes. A jointly written chapter [[59]] on the codes from finite geometries, for a handbook of coding theory edited by V. Pless and W. Huffman, is due to appear in 1998. This dealt exclusively with generalized Reed-Muller codes and their duals, and expanded the results described in Chapter 5 of the monograph [[51]]. The paper [[56]] updates some of the results and questions posed in [[51]] that were subsequently answered.

The most recent work includes a full classification of binary codes from Steiner triple systems [[54],[58]] and a paper defining a category associated with linear codes [[57]]. His last paper concerned binary codes associated with odd-order nets, and is joint with A. Drisko [[60]]. This was the work he spoke about in his excellent talk at Oberwolfach in March, the day before his death.

Over the years, his many papers show evidence of his many collaborations other than those mentioned above: he published jointly with most of his doctoral students, and with J. J. Florentin, M. T. Hermoso, J. -M. Goethals, V. Pless, and A. R. Prince.

Ed is survived by his wife Susan, daughter Alexi, and son Richard. He will be sorely missed by his friends and colleagues all over the world.

References

[[1]]

E. F. Assmus, Jr. On the homology of local rings. Illinois J. Math., 3:187-199, 1959.

[[2]]

E. F. Assmus, Jr. Betti numbers of local rings. Reports of the seminar on algebra Summer 1959, Technical Report No. 2, Contract NONR - 2121(12) between the office of Naval Research and the University of Chicago.

[[3]]

E. F. Assmus, Jr. A continuous function. Amer. Math. Monthly, 69:647, 1962.

[[4]]

E. F. Assmus, Jr. and H. F. Mattson. Error-correcting codes: an axiomatic approach. Inform. and Control, 6:315-330, 1963.

[[5]]

E. F. Assmus, Jr. On some mathematical aspects of Production from Consumption. Public Finance, XX:199-202, 1965.

[[6]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. Perfect codes and the Mathieu groups. Arch. Math., 17:122-135, 1966.

[[7]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. Disjoint Steiner systems associated with the Mathieu groups. Bull. Amer. Math. Soc., 72:843-845, 1966.

[[8]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. On the number of inequivalent Steiner triple systems. J. Combin. Theory, 1:301-305, 1966.

[[9]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. On tactical configurations and error-correcting codes. J. Combin. Theory, 2:243-257, 1967.

[[10]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. Intersections of cyclic Hamming codes. J. Combin. Theory, 3:307, 1967.

[[11]]

E. F. Assmus, Jr., H. F. Mattson, Jr., and R. J. Turyn. Research to Develop the Algebraic Theory of Codes. Applied Research Laboratory, Sylvania Electronic Systems, June 1967. No. AFCRL-67-0365. Contract No. AF19(628)-5998.

[[12]]

E. F. Assmus, Jr. and J. J. Florentin. Algebraic Machine Theory and Logical Design. Algebraic Theory of Machines, Chapter 2, pages 15-35, edited by Michael Arbib, Academic Press, 1968.

[[13]]

E. F. Assmus, Jr. On the optimality of cyclic codes. Proceedings of the Hawaii International Conference on System Sciences University of Hawaii Press, 1968.

[[14]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. Some (3p,p) codes. Information Processing 68, 1:205-209, 1969.

[[15]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. On the automorphism groups of Paley-Hadamard matrices. Combinatorial Mathematics and its Applications, Chapter 6, University of North Carolina Press, 1969.

[[16]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. New 5-designs. J. Combin. Theory, 6:122-151, 1969.

[[17]]

E. F. Assmus, Jr. The Projective Plane of Order 10? Mathematisches Forschungsinstitut, Oberwolfach, Germany, April 1, 1970

[[18]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. On the possibility of a projective plane of order 10. Algebraic Theory of Codes II. GTE Sylvania, October, 1970. AFCRL-71-0013. Contract No. F19628-69-C-0068.

[[19]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. On weights in quadratic-residue codes. Discrete Math., 3:1-20, 1972.

[[20]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. Contractions of self-orthogonal codes. Discrete Math., 3:21-32, 1972.

[[21]]

E. F. Assmus, Jr., H. F. Mattson, Jr. and Marcia Guza. Self-Orthogonal Steiner systems and projective planes. Math. Z., 138:89-96, 1974.

[[22]]

E. F. Assmus, Jr. and Maria Teresa Hermoso. Non-existence of Steiner systems of type S(d-1,d,2d). Math. Z., 138:171-172, 1974.

[[23]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. Coding and combinatorics. Siam Review, 16:349-388, 1974.

[[24]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. Some 3-error-correcting BCH codes have covering radius 5. IEEE Trans. Inform. Theory, 22:348-349, 1976.

[[25]]

E. F. Assmus, Jr., H. F. Mattson, Jr. and Howard E. Sachar. A new form of the square-root bound. SIAM J. Appl. Math., 30:352-354, 1976.

[[26]]

E. F. Assmus, Jr., J.-M. Goethals and H. F. Mattson, Jr. Generalized t-designs and majority decoding of linear codes. Information and Control, 32:43-60, 1976.

[[27]]

E. F. Assmus, Jr., Joseph A. Mezzaroba, and Chester J. Salwach. Planes and biplanes. In M. Aigner, editor, Higher Combinatorics, pages 205-212. D. Reidel, 1977. Proceedings of the NATO Conference, Berlin 1976.

[[28]]

E. F. Assmus, Jr. and Howard E. Sachar. Ovals from the point of view of coding theory. In M. Aigner, editor, Higher Combinatorics, pages 213-216. D. Reidel, 1977. Proceedings of the NATO Conference, Berlin 1976.

[[29]]

E. F. Assmus, Jr. and D. P. Maher. Nonexistence proofs for projective designs. Amer. Math. Monthly, 85:110-112, 1978.

[[30]]

E. F. Assmus, Jr. and H. F. Mattson, Jr. The weight-distribution of a coset of a linear code. IEEE Trans. Inform. Theory, IT-24:497, 1978.

[[31]]

E. F. Assmus, Jr. and Chester J. Salwach. The (16,6,2) designs. Internat. J. Math. & Math. Sci., 2:261-281, 1979.

[[32]]

E. F. Assmus, Jr. and J. H. van Lint. Ovals in projective designs. J. Combin. Theory, Ser. A, 27:307-324, 1979.

[[33]]

E. F. Assmus, Jr. and J. E. Novillo Sardi Generalized Steiner systems of type 3-(v, 4,6, 1). Proceedings of the Finite Geometries and Designs Conference Sussex June 1980, Cambridge University Press, 1981, 16-21.

[[34]]

E. F. Assmus, Jr. and Vera Pless. On the covering radius of extremal self-dual codes. IEEE Trans. Inform. Theory, IT-29:359-363, 1983.

[[35]]

E. F. Assmus, Jr. Applications of algebraic coding theory to finite geometric problems. In N. L. Johnson et al., editor, Finite Geometries, pages 23-32. New York: M. Dekker, 1983. Lecture Notes in Pure and Applied Math., 82.

[[36]]

E. F. Assmus, Jr. Estensioni per ovali di piani e bipiani e questioni collegate. Lecture Notes of lectures delivered in the Seminario di Geometrie Combinatorie diretto da G. Tallini, Universita Degli Studi di Roma, Instituto ``Guido Castelnuovo'', no. 45, Giugno 1983.

[[37]]

E. F. Assmus, Jr. The binary code arising from a 2-design with a nice collection of ovals. IEEE Trans. Inform. Theory, 29:367-369, 1983.

[[38]]

E. F. Assmus, Jr. Pi Amer. Math. Monthly, 92:213-214, 1985

[[39]]

E. F. Assmus, Jr. The non-existence of an oval-extendable (56, 11, 2) design. Journal of Geometry, 24:168-174, 1985

[[40]]

E. F. Assmus, Jr. Extending planar spaces. International Conference on Finite Geometry and Combinatorics, Deinze (Ghent), 1986.

[[41]]

E. F. Assmus, Jr. and J. D. Key. On an infinite class of Steiner systems with t = 3 and k = 6. J. Combin. Theory, Ser. A, 42:55-60, 1986.

[[42]]

E. F. Assmus, Jr. Biplanes in Planes. Congr. Numer., 60:279-294, 1987.

[[43]]

E. F. Assmus, Jr. The coding theory of finite geometries and designs. Lecture Notes in Computer Science 357:1-6, 1989.

[[44]]

E. F. Assmus, Jr. On the theory of designs. In J. Siemons, editor, Surveys in Combinatorics, 1989, pages 1-21. Cambridge: Cambridge University Press, 1989. London Mathematical Society Lecture Note Series 141.

[[45]]

E. F. Assmus, Jr. and J. D. Key. Arcs and ovals in the hermitian and Ree unitals. European J. Combin., 10:297-308, 1989.

[[46]]

E. F. Assmus, Jr. and J. D. Key. Affine and projective planes. Discrete Math., 83:161-187, 1990.

[[47]]

E. F. Assmus, Jr. and J. D. Key. Baer subplanes, ovals and unitals. In Dijen Ray-Chaudhuri, editor, Coding Theory and Design Theory, Part I, pages 1-8. New York: Springer-Verlag, 1990. The IMA Volumes in Mathematics and its Applications, Volume 20.

[[48]]

E. F. Assmus, Jr. and J. D. Key. Translation planes and derivation sets. J. Geom., 37:3-16, 1990.

[[49]]

E. F. Assmus, Jr. and Alan R. Prince. Biplanes and near biplanes. J. Geom., 40:1-14, 1991.

[[50]]

E. F. Assmus, Jr. On the Reed-Muller codes. Discrete Math., 106/107:25-33, 1992.

[[51]]

E. F. Assmus, Jr. and J. D. Key. Designs and their Codes. Cambridge University Press, 1992. Cambridge Tracts in Mathematics, Vol. 103 (Second printing with corrections, 1993).

[[52]]

E. F. Assmus, Jr. and J. D. Key. Hadamard matrices and their designs: a coding-theoretic approach. Trans. Amer. Math. Soc., 330:269-293, 1992.

[[53]]

Edward F. Assmus, Jr. and Jennifer D. Key. Codes and finite geometries. Technical report, INRIA, 1993. Report No. 2027.

[[54]]

E. F. Assmus, Jr. On 2-ranks of Steiner triple systems. Electron. J. Combin., 2:Research Paper 9, 1995.

[[55]]

E. F. Assmus, Jr. On Berman's characterization of the Reed-Muller codes. J. Statist. Plann. Inference, 56:17-21, 1996.

[[56]]

E. F. Assmus, Jr. and J. D. Key. Designs and codes: an update. Des. Codes Cryptogr., 9:7-27, 1996.

[[57]]

E. F. Assmus, Jr. The category of linear codes. IEEE Trans. Inform. Theory, 44:612-629, 1998.

[[58]]

E. F. Assmus, Jr. Linearly derived Steiner triple systems. Des. Codes Cryptogr., 13:1-19, 1998.

[[59]]

E. F. Assmus, Jr. and J. D. Key. Polynomial codes and finite geometries. Chapter in Handbook of Coding Theory, edited by V. Pless and W. C. Huffman. Elsevier. To appear 1998

[[60]]

E. F. Assmus, Jr. and A. Drisko. Binary codes of odd-order nets. Submitted.

Footnotes:

¹C. W. H. Lam, The search for a finite projective plane of order 10, Amer. Math. Monthly, 98 (1992) 305-318.