arXiv subjects
Publications and source records attributed to Xiang-dong Hou.
Let $f=a\x+\x^{3q-2}\in\Bbb F_{q^2}[\x]$, where $a\in\Bbb F_{q^2}^*$. We prove that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following occurs: (i) $q=2^e$, $e$ odd, and $a^{\frac{q+1}3}$ is a primitive $3$rd root of unity. (ii) $(q,a)$ belongs to a finite set which is determined in the paper.
Let $f=a{\tt x} +b{\tt x}^q+{\tt x}^{2q-1}\in\Bbb F_q[{\tt x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\Bbb F_{q^2}$. This result allows us to solve a related problem. Let $g_{n,q}\in\Bbb F_p[{\tt x}]$ ($n\ge 0$, $p=\text{char}\,\Bbb F_q$) be the polynomial defined by the functional equation $\sum_{c\in\Bbb F_q}({\tt x}+c)^n=g_{n,q}({\tt x}^q-{\tt x})$. We determine all $n$ of the form $n=q^α-q^β-1$, $α>β\ge 0$, for which $g_{n,q}$ is a permutation polynomial of $\Bbb F_{q^2}$.
Let $R$ be a commutative chain ring. We use a variation of Gröbner bases to study the lattice of ideals of $R[x]$. Let $I$ be a proper ideal of $R[x]$. We are interested in the following two questions: When is $R[x]/I$ Frobenius? When is $R[x]/I$ Frobenius and local? We develop algorithms for answering both questions. When the nilpotency of $\text{rad}\,R$ is small, the algorithms provide explicit answers to the questions.
Let $k$ be a positive integer and $S_{2k}={\tt x}+{\tt x}^4+...+{\tt x}^{4^{2k-1}}\in\Bbb F_2[{\tt x}]$. It was recently conjectured that ${\tt x}+S_{2k}^{4^{2k}}+S_{2k}^{4^k+3}$ is a permutation polynomial of $\Bbb F_{4^{3k}}$. In this note, the conjecture is confirmed and a generalization is obtained.
Let $q>2$ be a prime power and $f=-{\tt x}+t{\tt x}^q+{\tt x}^{2q-1}$, where $t\in\Bbb F_q^*$. We prove that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following occurs: (i) $q$ is even and $\text{Tr}_{q/2}(\frac 1t)=0$; (ii) $q\equiv 1\pmod 8$ and $t^2=-2$.
Let $q>2$ be a prime power and $f={\tt x}^{q-2}+t{\tt x}^{q^2-q-1}$, where $t\in\Bbb F_q^*$. It was recently conjectured that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following holds: (i) $t=1$, $q\equiv 1\pmod 4$; (ii) $t=-3$, $q\equiv \pm1\pmod{12}$; (iii) $t=3$, $q\equiv -1\pmod 6$. We confirm this conjecture in the present paper.
Let $p$ be a prime and $q$ a power of $p$. For $n\ge 0$, let $g_{n,q}\in\Bbb F_p[{\tt x}]$ be the polynomial defined by the functional equation $\sum_{a\in\Bbb F_q}({\tt x}+a)^n=g_{n,q}({\tt x}^q-{\tt x})$. When is $g_{n,q}$ a permutation polynomial (PP) of $\Bbb F_{q^e}$? This turns out to be a challenging question with remarkable breath and depth, as shown in the predecessor of the present paper. We call a triple of positive integers $(n,e;q)$ {\em desirable} if $g_{n,q}$ is a PP of $\Bbb F_{q^e}$. In the present paper, we find many new classes of desirable triples whose corresponding PPs were previously unknown. Several new techniques are introduced for proving a given polynomial is a PP.
For each pointed abelian group $(A,c)$, there is an associated {\em Galkin quandle} $G(A,c)$ which is an algebraic structure defined on $\Bbb Z_3\times A$ that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number of nonisomorphic pointed abelian groups of order $q^n$ ($q$ prime) is $\sum_{0\le m\le n}p(m)p(n-m)$, where $p(m)$ is the number of partitions of integer $m$.
A quandle is a self-distributive algebraic structure that appears in quasi-group and knot theories. For each abelian group A and c \in A we define a quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin quandles. Each G(A, c) is connected but not Latin unless A has odd order. G(A, c) is non-medial unless 3A = 0. We classify their isomorphism classes in terms of pointed abelian groups, and study their various properties. A family of symmetric connected quandles is constructed from Galkin quandles, and some aspects of knot colorings by Galkin quandles are also discussed.
Let $Λ=\Bbb Z[t,t^{-1}]$ be the ring of Laurent polynomials over $\Bbb Z$. We classify all $Λ$-modules $M$ with $|M|=p^n$, where $p$ is a primes and $n\le 4$. Consequently, we have a classification of Alexander quandles of order $p^n$ for $n\le 4$.
Let $D_n(1,x)$ be the $n$th reversed Dickson polynomial. The power sums $\sum_{a\in\Bbb F_q}D_n(1,a)^i$, $i=1,2$, have been determined recently. In this paper we give an evaluation of the sum $\sum_{a\in\Bbb F_q}D_n(1,a)^3$. This result implies new necessary conditions for $D_n(1,x)$ to be a permutation polynomial over $\Bbb F_q$.
Let $p$ be a prime and let $r, e, m$ be positive integers such that $r|e$ and $e|m$. The enumeration of linear codes of length $p^m$ over ${\Bbb F}_{p^r}$ which are invariant under the affine linear group ${\rm AGL}(\frac me, {\Bbb F}_{p^e})$ is equivalent to the enumeration of certain ideals in a partially ordered set $({\mathcal U}, \prec)$ where ${\mathcal U}=\{0,1,...,\frac me(p-1)\}^e$ and $\prec$ is defined by an $e$-dimensional simplicial cone. When $e=2$, the enumeration problem was solved in an earlier paper. In the present paper, we consider the cases $e=3$. We describe methods for enumerating all ${\rm AGL}(\frac m3, {\Bbb F}_{p^3})$-invariant linear codes of length $p^m$ over ${\Bbb F}_{p^r}$
We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the {\em vector degree}, are considered. We show that the number of irreducibles can be computed recursively by degree and that the number of relatively prime pairs can be expressed in terms of the number of irreducibles. We also obtain asymptotic formulas for the number of irreducibles and the number of relatively prime pairs. The asymptotic formulas for the number of irreducibles generalize and improve several previous results by Carlitz, Cohen and Bodin.
Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{α_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le α_1<...<α_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2πi\text{Tr}_n(y)/p}$, $y\in\Bbb F_{p^n}$, $\text{Tr}_n=\text{Tr}_{\Bbb F_{p^n}/\Bbb F_p}$. There is an effective way to compute the nullity of the quadratic form $\text{Tr}_{mn}(f(x))$ for all integer $m>0$. Assuming that all such nullities are known, we find relative formulas for $S(f,mn)$ in terms of $S(f,n)$ when $ν_p(m) \le \min\{ν_p(α_i):1\le i\le k\}$, where $ν_p$ is the $p$-adic order. We also find an explicit formula for $S(f,n)$ when $ν_2(α_1)=...= ν_2(α_k)<ν_2(n)$. These results generalize those by Carlitz and by Baumert and McEliece. Parallel results with $p=2$ were obtained in a previous paper by the second author.
Let $Ψ_{k,n}$ denote the number of inequivalent binary self-orthogonal $[n,k]$ codes. We present a method which allows us to compute $Ψ_{k,n}$ explicitly for a moderate $k$ and an arbitrary $n$. Included in this paper are explicit formulas for $Ψ_{k,n}$ with $k\le 5$.
For a permutation f of an n-dimensional vector space V over a finite field of order q we let k-affinity(f) denote the number of k-flats X of V such that f(X) is also a k-flat. By k-spectrum(n,q) we mean the set of integers k-affinity(f) where f runs through all permutations of V. The problem of the complete determination of k-spectrum(n,q) seems very difficult except for small or special values of the parameters. However, we are able to establish that k-spectrum(n,q) contains 0 in the following cases: (i) q>2 and 0 2. The maximum of k-affinity(f) is, of course, obtained when f is any semi-affine mapping. We conjecture that the next to largest value of k-affinity(f) is when f is a transposition and we are able to prove this when q=2, k=2, n>2 and when q>2, k=1, n>1.
Let $Ω$ be an algebraic closure of ${\mathbb Q}_p$ and let $F$ be a finite extension of ${\mathbb Q}_p$ contained in $Ω$. Given positive integers $f$ and $e$, the number of extensions $K/F$ contained in $Ω$ with residue degree $f$ and ramification index $e$ was computed by Krasner. This paper is concerned with the number ${\mathfrak I}(F,f,e)$ of $F$-isomorphism classes of such extensions. We determine ${\mathfrak I}(F,f,e)$ completely when $p^2\nmid e$ and get partial results when $p^2\parallel e$. When $s$ is large, ${\mathfrak I}({\mathbb Q}_p,f,e)$ is equal to the number of isomorphism classes of finite commutative chain rings with residue field ${\mathbb F}_{p^f}$, ramification index $e$, and length $s$.