10 problems

Chen et al.'s arithmetic autocorrelation distribution conjecture. The arithmetic autocorrelation satisfies

Let $\mathcal{M}$ be a binary $m$-sequence of period $2^n-1$, where $n\in\mathbb{N}$. For an integer shift $\tau$ with $1\leq\tau<2^n-1$, let $\mathcal{A}^{A}_{\mathcal{M}}(\tau)$…

Let $p$ be any odd prime, let $n$ be a positive integer, and set $N=p^n-1$. Let $\{s_t\}_{t=0}^{N-1}$ be the LSB sequence of a $p$-ary $m$-sequence of order $n$, and let $C_p$ deno…

Characterization conjecture. All three-valued cross correlations between two $m$-sequences of lengths $2^m-1$ and $2^{m/2}-1$ are characterized.

Let $r$ be a positive integer with $(r,3)=3$, and consider the cross correlation of a ternary $m$-sequence of period $3^{3r}-1$ with its decimation $d=3^r+2$ or $d=3^{2r}+2$. The c…

Let $p$ be a prime, let $n$ be a positive integer, and let $d$ be a decimation parameter. Define $\text{S}_d(z)=\text{C}_d(z)+1$, where $z\in\operatorname{GF}(p^n)^*$. Helleseth's…

Let $n=2t$ and $p=2$. Define the cross-correlation quantity $\text{S}_d(z)=\text{C}_d(z)+1$. Sarwate et al.'s conjecture. One has … The paper states that this conjecture is confirm…

Let $m=2n$, and let $d$ be a decimation satisfying … with the congruence understood as $d(2^l+1)=2^i\pmod{2^n-1}$, and suppose that $\gcd(l,n)=1$. The cross-correlation function…

Let $m>10$, and consider decimations $d$ in the cross correlation between an $m$-sequence of period $2^m-1$ and the $d$-decimation of an $m$-sequence of period $2^n-1$, where…

Let $m=2n$ be even, and let $d$ be a decimation defining the cross correlation between an $m$-sequence of period $2^m-1$ and the $d$-decimation of an $m$-sequence of period $2^n-1$…