本项目是本人理解格雷码与BCD编码之间转换算法之后写的一个Qt小程序。BCD转换到格雷码按照定义来就好，没有什么好优化的。但是格雷码到BCD码有多种算法可以讨论。

• $G_n$ is a permutation of the numbers 0, ..., $2^n − 1$. (Each number appears exactly once in the list.)
• $G_n$ is embedded as the first half of $G_n+1$.
• Therefore, the coding is stable, in the sense that once a binary number appears in $G_n$ it appears in the same position in all longer lists; so it makes sense to talk about the reflective Gray code value of a number:$G(m) $= the m-th reflecting Gray code, counting from 0.
• Each entry in $G_n$ differs by only one bit from the previous entry. (The Hamming distance is 1.)
• The last entry in $G_n$ differs by only one bit from the first entry. (The code is cyclic.)

the *$n$*th Gray code is obtained by computing ${\displaystyle n\oplus \lfloor n/2\rfloor }$

按照GrayCode的生成定义，并参照集成电路中移位寄存器，可以实现如下：

unsigned int BinaryToGray(unsigned int num)
{
    return num ^ (num >> 1);
}

根据GrayCode 的生成方式，GrayCode的第i位都包含BCD的第i位和第i-1位的信息（MSB，且可以看做移位寄存器初始为0），需要通过递归或者迭代的方式求出BCD编码， 下面是迭代的实现方式：

unsigned int GrayToBinary(unsigned int num)
{
    unsigned int mask = num >> 1;
    while (mask != 0)
    {
        num = num ^ mask;
        mask = mask >> 1;
    }
    return num;
}

在位数$n$为$2^k$时，可以使用分治的方式将运算次数降到$log_2n$从而减少运算量。同时位数$n$不为$2^k$时，也可以高位补零到最近的$2^k$从而减少计算量。

unsigned int GrayToBinary32(unsigned int num)
{
    num = num ^ (num >> 16);
    num = num ^ (num >> 8);
    num = num ^ (num >> 4);
    num = num ^ (num >> 2);
    num = num ^ (num >> 1);
    return num;
}