Session S29 - Theory and Applications of Coding Theory

Tuesday, July 13, 15:00 ~ 15:25 UTC-3

On the number of solutions of systems of certain diagonal equations over finite fields.

Melina Privitelli

Universidad Nacional de General Sarmiento / CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $\mathbb{F}_{\hskip-0.7mm q}$ be the finite field of $q$ elements. It is a classical problem to determine or to estimate the number $N$ of $\mathbb{F}_{\hskip-0.7mm q}$--rational solutions (i.e. solutions with coordinates in $\mathbb{F}_{\hskip-0.7mm q}$) of systems of polynomial equations over $\mathbb{F}_{\hskip-0.7mm q}$ (see, e.g., [5]). There are explicit formulas for the number $N$ only for some very particular cases (see, e.g., [1] and [8]) . For this reason, it is important to have estimates on the number $N$ and results which guarantee the existence of this kind of solutions.

Particularly, the systems of diagonal equations \begin{equation}(1)\hspace{1in}\left \{\begin{array}{ccl} a_{11}X_1^{d_1} & + a_{12}X_2^{d_2} + \cdots + & a_{1t}X_t^{d_t} = b_1 \\ a_{21}X_1^{d_1} & + a_{22}X_2^{d_2} + \cdots + & a_{2t}X_t^{d_t} = b_2 \\ \;\vdots & &\quad\vdots \\ a_{n1}X_1^{d_1} &+ a_{n2}X_2^{d_2} + \cdots +& a_{nt}X_t^{d_t} = b_n, \end{array}\right. \end{equation} with $b_1,\ldots,b_n\in \mathbb{F}_{\hskip-0.7mm q}$, have been considered in the literature because the study of its set of $\mathbb{F}_{\hskip-0.7mm q}$--rational solutions has several applications to different areas of mathematics, such as the theory of cyclotomy, Waring’s problem and the graph coloring problem (see, e.g. [3] and [5]). Additionally, information on the number $N$ is very useful in different aspects of coding theory such as the weight distribution of some cyclic codes ([9] and [10]) and the covering radius of certain cyclic codes ( [2] and [4]).

In comparison with a single diagonal equation, there are much fewer results about the number of $\mathbb{F}_{\hskip-0.7mm q}$-rational solutions of systems of the type (1) and most of them use tools involving character sums. In this work, we approach this problem using tools of algebraic geometry. More precisely, we consider an $\mathbb{F}_{\hskip-0.7mm q}$-variety $V$ associated to the system. We study the geometric properties of $V$, where the key point is obtaining upper bounds of the dimension of its singular locus. This study allows us to obtain estimates and existence results of rational solutions of systems of diagonal equations which in particular improve W. Spackman’s (see [6] and [7]). Furthermore, our techniques can be applied to the study of some variants of these systems such as systems of Dickson’s equations and generalized diagonal equations.

References.

[1] X. Cao, W-S. Chou and J. Gu, On the number of solutions of certain diagonal equations over finite fields, Finite Fields Appl. 42 (2016), 225--252.

[2] T. Helleseth, On the covering radius of cyclic linear codes and arithmetic codes, Discrete Appl. Math. 11(1985), no. 2, 157--173.

[3] R. Lidl and H. Niederreiter, Finite fields, Addison--Wesley, Reading, Massachusetts, 1983.

[4] O. Moreno and F. N. Castro, Divisibility properties for covering radius of certain cyclic codes, IEEE Trans. Inform. Theory 49 (2003), no. 12, 3299--3303.

[5] Gary L. Mullen and Daniel Panario, Handbook of Finite Fields (1st ed.), Chapman and Hall/CRC, 2013.

[6] K. W. Spackman, Simultaneous solutions to diagonal equations over finite fields, J. Number Theory 11(1979), no. 1, 100--115.

[7] K. W. Spackman, On the number and distribution of simultaneous solutions to diagonal congruences, Canadian J. Math. 33 (1981), no. 2, 421--436.

[8] J. Wolfmann, Some systems of diagonal equations over finite fields, Finite Fields Appl. 4(1998), no. 1, 29--37.

[9] X. Zeng L. Hu, W. Jiang, Q. Yue and X. Cao, The weight distribution of a class of $p$-ary cyclic codes, Finite Fields Appl. 16 (2010), no.1, 56--73.

[10] D. Zheng, X. Wang, X. Zeng and L. Hu, The weight distribution of a family of $p$-ary cyclic codes, Des. Codes Cryptogr. 75(2015), no. 2, 263--275.

Joint work with Mariana Pérez, Universidad Nacional de Hurlingham, CONICET.

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