The page reproduced above is from Gauss,
and concerns what we now call Gauss sums,
which crop up often in my work.

Daniel J. Katz

Assistant Professor
Department of Mathematics
California State University, Northridge

I investigate problems in number theory and discrete
mathematics, often motivated by information theory.
Both algebra and analysis play crucial roles.

Some of my recent work includes a proof for finite fields
of characteristic 2 and 3 of a conjecture of Helleseth (1976)
concerning cross-correlations of maximal linear recursive
sequences (equivalent to a conjecture about Weil sums of
binomials or a statement about weights in certain error-
correcting codes).

In a similar area, Philippe Langevin and I proved a conjecture
of Dobbertin, Helleseth, Kumar, and Martinsen (2001) that
asserts the existence of an infinite a three-valued family of
Weil sums of binomials. This is the tenth infinite family
discovered since 1966.

Jonathan Jedwab, Kai-Uwe Schmidt, and I have settled con-
jectures of Høholdt-Jensen (1988), Borwein-Choi-Jedwab (2004),
Parker (2005), Yu-Gong (2007), and Jedwab-Schmidt (2010) on
the asymptotic L4 norm of certain families of Littlewood
polynomials. In doing so, we break a record (which stood for
over two decades) for the lowest known asymptotic mean-square
autocorrelation for binary sequences.

Seminar

I organize the CSUN Algebra, Number Theory, and Discrete Math Seminar

Symposium

I organized the Seventeenth Annual Sigma Xi Student Research Symposium of the CSUN Chapter of Sigma Xi

Mathematics Papers

  1. (with P. Langevin, S. Lee, and Y. Sapozhnikov) Valuations of Weil Sums of Binomials
          arXiv: 1608.04047 [math.NT]

  2. (with T. Boothby) Low Correlation Sequences from Linear Combinations of Characters
          arXiv: 1602.04514 [cs.IT]

  3. Aperiodic Crosscorrelation of Sequences Derived from Characters
         IEEE Transactions on Information Theory, 62(9): 5237-5259 (2016).
          arXiv: 1602.04487 [cs.IT]

  4. (with R. A. Cowan and L. M. White) A New Generating Function for Calculating the Igusa Local Zeta Function
          Advances in Mathematics, 304: 355-420 (2017).
          arXiv: 1506.07869 [math.NT]

  5. (with B. Abrego, S. Fernandez-Merchant, and L. Kolesnikov) On The Number of Similar Instances of a Pattern in a Finite Set
          arXiv: 1501.00076 [math.CO]

  6. (with P. Langevin) New Open Problems Related to Old Conjectures by Helleseth
          Cryptography and Communications, 8(2): 175-189 (2016).
          arXiv: 1412.8530 [math.NT]

  7. (with P. Langevin) Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials
          Acta Arithmetica, 169(2): 181-199 (2015).
          arXiv: 1409.2459 [math.NT]

  8. Divisibility of Weil Sums of Binomials
          Proceedings of the American Mathematical Society 143(11): 4623-4632 (2015).
          arXiv: 1407.7923 [math.NT]

  9. (with Y. Aubry and P. Langevin) Cyclotomy of Weil Sums of Binomials
          Journal of Number Theory, 154: 160-178 (2015).
          arXiv: 1312.3889 [math.NT]
  10.       Summary in French: Cyclotomie des sommes de Weil binomiales
          Comptes Rendus Mathématique. Académie des Sciences. Paris, 352(5): 373-376 (2014).

  11. (with J. Jedwab and K.-U. Schmidt) Littlewood Polynomials with Small L4 Norm
          Advances in Mathematics, 241: 126-136 (2013).
          arXiv: 1205.0260 [math.NT]

  12. (with J. Jedwab and K.-U. Schmidt) Advances in the Merit Factor Problem for Binary Sequences
          Journal of Combinatorial Theory Series A, 120(4): 882-906 (2013).
          arXiv: 1205.0626 [math.CO]

  13. Asymptotic L4 Norm of Polynomials Derived from Characters
          Pacific Journal of Mathematics, 263(2): 373-398 (2013).
          arXiv:1205.1069 [math.NT]

  14. Weil Sums of Binomials, Three-Level Cross-Correlation, and a Conjecture of Helleseth
          Journal of Combinatorial Theory Series A, 119(8): 1644-1659 (2012).
          (one result in the paper was announced beforehand in arXiv:1105.2291v1 [math.CO])

  15. On Theorems of Delsarte-McEliece and Chevalley-Warning-Ax-Katz
          Designs, Codes and Cryptography, 65(3): 291--324 (2012).

  16. Point Count Divisibility for Algebraic Sets over Z/plZ and Other Finite Principal Rings
          Proceedings of the American Mathematical Society, 137(12): 4065-4075 (2009).

  17. Sharp p-Divisibility of Weights in Abelian Codes over Z/pdZ
          IEEE Transactions on Information Theory, 54(12): 5354-5380 (2008).
          with a correction

  18. (with J. Zahl) Bounds on Degrees of p-Adic Separating Polynomials
         Journal of Combinatorial Theory Series A, 115(7): 1310-1319 (2008).

  19. p-Adic Estimates of Hamming Weights in Abelian Codes over Galois Rings
         IEEE Transactions on Information Theory, 52(3): 964-985 (2006).

  20. p-Adic Valuation of Weights in Abelian Codes over Zpd
         IEEE Transactions on Information Theory, 51(1): 281-305 (2005).

Contact Information

      Department of Mathematics
      California State University, Northridge
      18111 Nordhoff Street
      Northridge, CA 91328-8313
      USA

      telephone: (818) 677-2712

      email: [my first name] [dot] [my last name] [at] csun [dot] edu