论文标题
在三元素代码集中的前缀代码的比例
On the proportion of prefix codes in the set of three-element codes
论文作者
论文摘要
令$ l $为自然数的有限顺序。在Woryna（2017,2018）中，我们得出了比率$ρ_{n，l} = | pr_n（l）|/| ud_n（l）| $的一些有趣属性前缀代码的相应子集。在本文中，我们研究长度分布是三元素序列的情况。在这种情况下，我们表明比率$ρ_{n，l} $总是大于$α_n$，其中$α_n=（n-2）/n $ for $ n> 2 $和$α_2= 1/6 $。此外，数字$α_n$是该比率的最佳下限，因为表格$ l =（1,1，c）$和$ l =（1,2，c）的长度分布确保该比率渐近接近$α_n$。也就是说，如果$ l =（1,1，c）$，则$ρ_{n，l} $倾向于$（n-2）/n $，带有$ c \ to \ to \ infty $，并且，如果$ l =（1,2，c）$，则$ρ_{2，l}倾向于$ 1/6 $ a $ c with $ c with $ c \ c \ f infty $ \ f infty $ \ f infty $ \ f infty $ \ f ty \ f ty \ f ty \ f infty $。
Let $L$ be a finite sequence of natural numbers. In Woryna (2017,2018), we derived some interesting properties for the ratio $ρ_{n,L}=|PR_n(L)|/|UD_n(L)|$, where $UD_n(L)$ denotes the set of all codes over an $n$-letter alphabet and with length distribution $L$, and $PR_n(L)\subseteq UD_n(L)$ is the corresponding subset of prefix codes. In the present paper, we study the case when the length distributions are three-element sequences. We show in this case that the ratio $ρ_{n,L}$ is always greater than $α_n$, where $α_n=(n-2)/n$ for $n>2$ and $α_2=1/6$. Moreover, the number $α_n$ is the best possible lower bound for this ratio, as the length distributions of the form $L=(1,1,c)$ and $L=(1,2,c)$ assure that the ratios asymptotically approach $α_n$. Namely, if $L=(1,1,c)$, then $ρ_{n,L}$ tends to $(n-2)/n$ with $c\to\infty$, and, if $L=(1,2,c)$, then $ρ_{2,L}$ tends to $1/6$ with $c\to\infty$.
