The Sierpiński Triangle over Finite Fields

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\title{The Sierpi\'nski Triangle over Finite Fields}

\author{
    {\large Sara D. Cardell}\thanks{[email protected]} \\
    {\small Unesp, Rio Claro, SP} \\
   {\large Miguel Beltrá}\thanks{[email protected]}, {\large Verónica Requena}\thanks{[email protected]}  \\
    {\small University of Alicante, Alicante, Spain} \\
}
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 The binary binomial sequences  correspond to the diagonals of the binary Pascal's triangle (see Figure~\ref{fig:Pascal2}).
They have interesting properties, such as all binary sequences with a period power of $2$ can be computed as the XOR of a finite set of binomial sequences \cite{Cardell2019c}.
Other properties of these sequences (period, linear complexity, construction rules, or relations among them)  have been deeply analyzed for the binary case \cite{Cardell2019c}.

 \begin{figure}[h]
     \centering
     \input{triang2}
     \caption{Pascal's triangle modulo 2. Source: the authors.}
     \label{fig:Pascal2}
 \end{figure}

Let $p$ be a prime number, $\mathbb{F}_p$ be the Galois field of $p$ elements,, and $k\geq 0$ a fixed integer.
% We say $\{a_n\}_{n\geq 0}=\{a_0, a_1, a_2, \ldots\}$ is a \textbf{$p-$ary sequence} if its terms $a_n\in \mathbb{F}_p$,
% for $n\geq 0$. The sequence $\{a_n\}_{n\geq 0}$ is \textbf{periodic} if and only if there exists an integer $T$ such that $a_{n+T}=a_n$, for all $n\geq 0$.
% In the sequel, all the sequences considered will be $p-$ary sequences and  the term $+$ will denote  the sum modulo $p$.
The \textbf{($p-$ary) $k-$th binomial sequence} $\left\{\binom{n}{k}\right\}_{n\geq 0} $  is 
the sequence whose terms are given by the binomial coefficients modulo $p$, that is,
$\binom{n}{k} \bmod p$,   if   $n\geq k$ and      $0$ if  $n< k$. 
These sequences correspond to the diagonals of Pascal's triangle modulo $p$.
For instance, in Figure~\ref{fig:pascal}, Pascal's triangle modulo 3 is displayed, revealing the fractal structure of the sequences and their underlying patterns.

In this work, we study the construction rules of these sequences and other properties of this family of sequences over $\mathbb{F}_p$. 
For example, it is  possible to prove that the linear complexity of the sequence 
 $\left\{\binom{n}{k}\right\}_{n\geq 0}$ is $LC=k+1$ and the characteristic polynomial of the sequence is $p(x)=(x-1)^{k+1}$.
 Furthermore, we show that any $p$-ary sequence $\{ a_n \} = \{a_0,a_1,a_2,\ldots,a_{p^L-1}, \ldots\}$ of period $T = p^L$, with $L>0$ an integer, can be written as a linear combination of $p$-ary binomial sequences. 
That is, there exist $\alpha_i \in \mathbb{F}_p$, with $i \in \{0,1,\ldots,p^L-1\}$,
%$c_0,c_1,c_2,\ldots,c_{p^L-1}$ 
such that
\begin{equation} \label{eq:binomial}
\{ a_n \}
=
\sum_{i=0}^{p^L-1}\alpha_i
\left\{\binom{n}{i}\right\}_{n\geq 0}.
\end{equation}
The expression \eqref{eq:binomial} is called the \textbf{binomial representation} of $\{a_n \}_{n\geq 0}$. 
It is possible to deduce properties of the sequence $\{a_n \}_{n\geq 0}$ simply by observing its binomial representation.
 

\begin{figure}
    \centering
    \input{triang3}
    \caption{Pascal's triangle modulo 3. Source: the authors.}
    \label{fig:pascal}
\end{figure}

\section*{Acknowledgments}
The first author was supported by CNPq with process 405842/2023-6 and by FAPESP with process 2024/05051-
7.
The work of the second and third author was supported by I+D+i projects VIGROB23-
287 and UADIF23-132 of the Universitat d’Alacant. 

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