first page of Gauss's Summatio quarumdam serierum singularium


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

Associate Professor
Department of Mathematics
California State University, Northridge

I investigate problems in number theory and discrete
mathematics, often motivated by information theory.

Some of my recent work includes a proof for finite fields
of characteristic 2 and 3 of a conjecture of Helleseth (1971)
concerning crosscorrelations of maximal linear recursive
sequences (or equivalently, about Weil sums of binomials
binomials, nonlinearity of finite field permutations, or
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.

Tor Helleseth, Chunlei Li, and I settled the final part of the
last conjecture in Niho's thesis (1972), which also concerns
Weil sums that determine crosscorrelations, nonlinearity,
and weights in in codes.

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

Papers

  1. (with Courtney van der Linden) Peak Sidelobe Level and Peak Crosscorrelation of Golay-Rudin-Shapiro Sequences
          arXiv: 2108.07318 [cs.IT]

  2. (with T. Helleseth and C. Li) The resolution of Niho's last conjecture concerning sequences, codes, and Boolean functions
          to appear in IEEE Transactions on Information Theory.
          arXiv: 2006.12239 [math.NT] (Sage code with verification of decomposition in Lemma 5.2 included here)

  3. (with S. R. Garcia and G. Karaali) An improved uncertainty principle for functions with symmetry
         Journal of Algebra, 586: 899-934 (2021).
          arXiv: 1807.07648 [math.CA]

  4. Sequences with Low Correlation
          Arithmetic of Finite Fields, 7th International Workshop, WAIFI 2018, Bergen, Norway, June 14-16, 2018, Revised Selected Papers,
          volume 11321 of Lecture Notes in Computer Science, 149-172 (2018).
          arXiv: 1806.04707 [cs.IT]

  5. Weil sums of binomials: properties, applications and open problems
          Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications,
          De Gruyter, Berlin, Boston, pp. 109-134 (2019).
          arXiv: 1805.10452 [math.NT]

  6. (with S. Lee and S. A. Trunov) Rudin-Shapiro-Like Sequences With Maximum Asymptotic Merit Factor
         IEEE Transactions on Information Theory, 66(12): 7728-7738 (2020).
          arXiv: 1711.02233 [cs.IT]

  7. (with E. Moore) Sequence Pairs with Lowest Combined Autocorrelation and Crosscorrelation
          arXiv: 1711.02229 [cs.IT]

  8. (with S. Lee and S. A. Trunov) Crosscorrelation of Rudin-Shapiro-Like Polynomials
         Applied and Computational Harmonic Analysis, 48(2): 513-538 (2020).
          arXiv: 1702.07697 [cs.IT] (raw data can be found here)

  9. (with P. Langevin, S. Lee, and Y. Sapozhnikov) The p-Adic Valuations of Weil Sums of Binomials
         Journal of Number Theory, 181: 1-26 (2017).
          arXiv: 1608.04047 [math.NT] (raw data can be found here)

  10. (with K. T. R. Boothby) Low Correlation Sequences from Linear Combinations of Characters
         IEEE Transactions on Information Theory, 63(10): 6158-6178 (2017).
          arXiv: 1602.04514 [cs.IT] (raw data can be found here)

  11. Aperiodic Crosscorrelation of Sequences Derived from Characters
         IEEE Transactions on Information Theory, 62(9): 5237-5259 (2016).
          arXiv: 1602.04487 [cs.IT] (raw data can be found here)

  12. (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]

  13. (with B. Abrego, S. Fernandez-Merchant, and L. Kolesnikov) On The Number of Similar Instances of a Pattern in a Finite Set
          Electronic Journal of Combinatorics, 23(4): P4.39 (2016).
          arXiv: 1501.00076 [math.CO]

  14. (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]

  15. (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]

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

  17. (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]
  18.       Summary in French: Cyclotomie des sommes de Weil binomiales
          Comptes Rendus Mathématique. Académie des Sciences. Paris, 352(5): 373-376 (2014).

  19. (with J. Jedwab and K.-U. Schmidt) Littlewood Polynomials with Small L4 Norm
          Advances in Mathematics, 241: 127-136 (2013).
          arXiv: 1205.0260 [math.NT]

  20. (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]

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

  22. 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])

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

  24. 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).

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

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

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

  28. 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