Gopala-Hemachandra codes are a variation of the Fibonacci universal code and have applications in cryptography and data compression. We show that $GH_{a}(n)$ codes always exist for $a=-2,-3$ and $-4$ for any integer $n \geq 1$ and hence are universal codes. We develop two new algorithms to determine whether a GH code exists for a given set of parameters $a$ and $n$. In 2010, Basu and Prasad showed experimentally that in the range $1 \leq n \leq 100$ and $1 \leq k \leq 16$, there are at most $k$ consecutive integers for which $GH_{-(4+k)}(n)$ does not exist. We turn their numerical result into a mathematical theorem and show that it is valid well beyond the limited range considered by them.

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重新审视 Gopala-Hemachandra 代码

Gopala-Hemachandra 码是斐波那契通用码的变体，在密码学和数据压缩方面有应用。我们证明 $GH_{a}(n)$ 代码总是存在于 $a=-2,-3$ 和 $-4$ 对于任何整数 $n \geq 1$ 并且因此是通用代码。我们开发了两种新算法来确定给定参数 $a$ 和 $n$ 的 GH 代码是否存在。2010 年，Basu 和 Prasad 实验表明，在 $1 \leq n \leq 100$ 和 $1 \leq k \leq 16$ 范围内，至多有 $k$ 个连续整数满足 $GH_{-(4+k) }(n)$ 不存在。我们将他们的数值结果转化为数学定理，并表明它在超出他们考虑的有限范围内是有效的。