几何分布随机变量对的最优前缀码
Optimal prefix codes for pairs of geometrically-distributed random variables
摘要：
研究了由具有参数$q$，$0{<}q{<}1$的几何概率分布的源发射的独立整数值符号对的最佳前缀码。通过对符号对进行编码，可以减少逐符号编码的冗余代价，同时保持Golomb码及其变体的典型编码和解码过程的简单性。它示出，这些所谓的二维几何分布的最佳代码是\emph{singular}，在这个意义上说，一个前缀码是最佳的一个值的参数$q$不能是最佳的任何其他值的$q$。这与一维情况形成鲜明对比，在一维情况下，对于参数$q$的正长度间隔，代码是最佳的。因此，在二维的情况下，它是不可行的，以给出一个紧凑的特性的最佳代码的所有值的参数$q$，因为是在一维的情况下。相反，最优码的特征在于$q$的值的离散序列，其提供单位间隔的良好覆盖。具体地，针对覆盖范围$q\ge 1/2$的$q=2^{-1/k}$（$k\ge 1$）和覆盖范围$q<1/2$的$q=2^{-k}$（$k>1$）描述最优前缀码。所描述的代码相对于一维情况产生预期的冗余减少，同时保持低复杂度的编码操作。
英文摘要：
Optimal prefix codes are studied for pairs of independent, integer-valued symbols emitted by a source with a geometric probability distribution of parameter $q$, $0{<}q{<}1$. By encoding pairs of symbols, it is possible to reduce the redundancy penalty of symbol-by-symbol encoding, while preserving the simplicity of the encoding and decoding procedures typical of Golomb codes and their variants. It is shown that optimal codes for these so-called two-dimensional geometric distributions are \emph{singular}, in the sense that a prefix code that is optimal for one value of the parameter $q$ cannot be optimal for any other value of $q$. This is in sharp contrast to the one-dimensional case, where codes are optimal for positive-length intervals of the parameter $q$. Thus, in the two-dimensional case, it is infeasible to give a compact characterization of optimal codes for all values of the parameter $q$, as was done in the one-dimensional case. Instead, optimal codes are characterized for a discrete sequence of values of $q$ that provide good coverage of the unit interval. Specifically, optimal prefix codes are described for $q=2^{-1/k}$ ($k\ge 1$), covering the range $q\ge 1/2$, and $q=2^{-k}$ ($k>1$), covering the range $q<1/2$. The described codes produce the expected reduction in redundancy with respect to the one-dimensional case, while maintaining low complexity coding operations.
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