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Zhixiong Chen

Publications and source records attributed to Zhixiong Chen.

44 records · Page 3Linked to original sources
arXiv2015

Linear complexity and trace representation of quaternary sequences over $\mathbb{Z}_4$ based on generalized cyclotomic classes modulo $pq$

We define a family of quaternary sequences over the residue class ring modulo $4$ of length $pq$, a product of two distinct odd primes, using the generalized cyclotomic classes modulo $pq$ and calculate the discrete Fourier transform (DFT) of the sequences. The DFT helps us to determine the exact values of linear complexity and the trace representation of the sequences.

math.NT↗
arXiv2014

Linear complexity problems of level sequences of Euler quotients and their related binary sequences

The Euler quotient modulo an odd-prime power $p^r~(r>1)$ can be uniquely decomposed as a $p$-adic number of the form $$ \frac{u^{(p-1)p^{r-1}} -1}{p^r}\equiv a_0(u)+a_1(u)p+\ldots+a_{r-1}(u)p^{r-1} \pmod {p^r},~ \gcd(u,p)=1, $$ where $0\le a_j(u) 1$. We firstly study certain arithmetic properties of the level sequences $(a_j(u))_{u\ge 0}$ over $\mathbb{F}_p$ via introducing a new quotient. Then we determine the exact values of linear complexity of $(a_j(u))_{u\ge 0}$ and values of $k$-error linear complexity for binary sequences defined by $(a_j(u))_{u\ge 0}$.

math.NT↗
arXiv2014

Polynomial quotients: Interpolation, value sets and Waring's problem

For an odd prime $p$ and an integer $w\ge 1$, polynomial quotients $q_{p,w}(u)$ are defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~~ \mathrm{with}~~ 0 \le q_{p,w}(u) \le p-1, ~~u\ge 0, $$ which are generalizations of Fermat quotients $q_{p,p-1}(u)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f(u)\equiv q_{p,w}(u) \bmod p$ for a given polynomial $f(x)$ over the finite field $\mathbb{F}_p$. In particular, for the case $f(x)=x$ we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each element of $\mathbb{F}_p$ as sum of values of polynomial quotients, we prove some lower bounds on the size of their value sets, and then we apply these lower bounds to prove some bounds on the Waring number using results from bounds on additive character sums and additive number theory.

math.NT↗
arXiv2013

Covering sets for limited-magnitude errors

For a set $\cM=\{-μ,-μ+1,\ldots, λ\}\setminus\{0\}$ with non-negative integers $λ,μ<q$ not both 0, a subset $\cS$ of the residue class ring $\Z_q$ modulo an integer $q\ge 1$ is called a $(λ,μ;q)$-\emph{covering set} if $$ \cM \cS=\{ms \bmod q : m\in \cM,\ s\in \cS\}=\Z_q. $$ Small covering sets play an important role in codes correcting limited-magnitude errors. We give an explicit construction of a $(λ,μ;q)$-covering set $\cS$ which is of the size $q^{1 + o(1)}\max\{λ,μ\}^{-1/2}$ for almost all integers $q\ge 1$ and of optimal size $p\max\{λ,μ\}^{-1}$ if $q=p$ is prime. Furthermore, using a bound on the fourth moment of character sums of Cochrane and Shi we prove the bound $$ω_{λ,μ}(q)\le q^{1+o(1)}\max\{λ,μ\}^{-1/2},$$ for any integer $q\ge 1$, however the proof of this bound is not constructive.

cs.IT↗
arXiv2013

On the $k$-error linear complexity of binary sequences derived from polynomial quotients

We investigate the $k$-error linear complexity of $p^2$-periodic binary sequences defined from the polynomial quotients (including the well-studied Fermat quotients), which is defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~ \mathrm{with} 0 \le q_{p,w}(u) \le p-1, ~u\ge 0, $$ where $p$ is an odd prime and $1\le w<p$. Indeed, first for all integers $k$, we determine exact values of the $k$-error linear complexity over the finite field $\F_2$ for these binary sequences under the assumption of f2 being a primitive root modulo $p^2$, and then we determine their $k$-error linear complexity over the finite field $\F_p$ for either $0\le k<p$ when $w=1$ or $0\le k<p-1$ when $2\le w<p$. Theoretical results obtained indicate that such sequences possess `good' error linear complexity.

cs.CR↗
arXiv2013

Trace representation and linear complexity of binary sequences derived from Fermat quotients

We describe the trace representations of two families of binary sequences derived from Fermat quotients modulo an odd prime $p$ (one is the binary threshold sequences, the other is the Legendre-Fermat quotient sequences) via determining the defining pairs of all binary characteristic sequences of cosets, which coincide with the sets of pre-images modulo $p^2$ of each fixed value of Fermat quotients. From the defining pairs, we can obtain an earlier result of linear complexity for the binary threshold sequences and a new result of linear complexity for the Legendre-Fermat quotient sequences under the assumption of $2^{p-1}\not\equiv 1 \bmod {p^2}$.

math.NT↗