> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/data-structures/gray-code.md).

# 格雷编码

### 定义

格雷编码是一种相邻两个数**只有一位不同**的编码方式，常用于减少数字电路切换错误。

### 二进制与格雷码的转换

* **二进制转格雷码**：`g = b ^ (b >> 1)`
* **格雷码转二进制**：最高位相同，其后每位等于前一位二进制与当前格雷码位的异或

```cpp
// 二进制 n 转格雷码
int toGray(int n) { return n ^ (n >> 1); }

// 格雷码 g 转二进制
int toBinary(int g) {
    int b = g;
    while (g >>= 1) b ^= g;
    return b;
}
```

### 3 位格雷码

```
000, 001, 011, 010, 110, 111, 101, 100
```

相邻两数（含首尾）都只差一位，故格雷码也是一种“循环码”。

### 性质

n 位格雷码共有 `2^n` 个码字，其中第 i 个格雷码就是 `i ^ (i >> 1)`。


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