The page reproduced above is from Gauss, and concerns what we now call Gauss sums, which crop up often in my work. |
## Daniel J. KatzAssistant ProfessorDepartment 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 L ^{4} norm of certain families of Littlewoodpolynomials. In doing so, we break a record (which stood for over two decades) for the lowest known asymptotic mean-square autocorrelation for binary sequences. |

- (with S. R. Garcia and G. Karaali) On Chebotarëv's nonvanishing minors theorem and the Biró-Meshulam-Tao discrete uncertainty principle

arXiv: 1807.07648 [math.CA] - 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] - Weil sums of binomials: properties and applications

arXiv: 1805.10452 [math.NT] - (with S. Lee and S. A. Trunov) Rudin-Shapiro-Like Polynomials with Maximum Asymptotic Merit Factor

arXiv: 1711.02233 [cs.IT] - (with E. Moore) Sequence Pairs with Lowest Combined Autocorrelation and Crosscorrelation

arXiv: 1711.02229 [cs.IT] - (with S. Lee and S. A. Trunov) Crosscorrelation of Rudin-Shapiro-Like Polynomials

*Applied and Computational Harmonic Analysis*, in press.

arXiv: 1702.07697 [cs.IT] - (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) - (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) - 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) - (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] - (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] - (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] - (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] - Divisibility of Weil Sums of Binomials

*Proceedings of the American Mathematical Society*143(11): 4623-4632 (2015).

arXiv: 1407.7923 [math.NT] - (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]
Summary in French: Cyclotomie des sommes de Weil binomiales - (with J. Jedwab and K.-U. Schmidt) Littlewood Polynomials with Small L
^{4}Norm

*Advances in Mathematics*, 241: 127-136 (2013).

arXiv: 1205.0260 [math.NT] - (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] - Asymptotic L
^{4}Norm of Polynomials Derived from Characters

*Pacific Journal of Mathematics*, 263(2): 373-398 (2013).

arXiv:1205.1069 [math.NT] - 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]) - On Theorems of Delsarte-McEliece and Chevalley-Warning-Ax-Katz

*Designs, Codes and Cryptography*, 65(3): 291--324 (2012). - Point Count Divisibility for Algebraic Sets over Z/p
^{l}Z and Other Finite Principal Rings

*Proceedings of the American Mathematical Society*, 137(12): 4065-4075 (2009). - Sharp p-Divisibility of Weights in Abelian Codes over Z/p
^{d}Z

*IEEE Transactions on Information Theory*, 54(12): 5354-5380 (2008).

with a correction - (with J. Zahl) Bounds on Degrees of p-Adic Separating Polynomials

*Journal of Combinatorial Theory Series A*, 115(7): 1310-1319 (2008). - p-Adic Estimates of Hamming Weights in Abelian Codes over Galois Rings

*IEEE Transactions on Information Theory*, 52(3): 964-985 (2006). - p-Adic Valuation of Weights in Abelian Codes over Z
_{pd}

*IEEE Transactions on Information Theory*, 51(1): 281-305 (2005).

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