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第二章:信道容量和纠错2 ^BXJSCjow
对于转移概率小于0.1(一般都是),只需要使用固定的
汉明码即可。 ^6G27Y9gy
一般只会考察(7,4)(6,3)汉明码所以直接记住就好啦。 ^mcTAsu6l
(7,4)汉明码 G 矩阵维度:4×7(4 行 7 列) ^YojZjC7f
确定(7,4)的 n 和 k: ^ap8vx34k
n=7(码字总位数:4 位信息位 + 3 位校验位,r=7-4=3); ^5gIMRzO7
k=4(信息位位数:原始消息是 4 位二进制数,如 0001、1010)。 ^lbM3qnMR
文档中(7,4)汉明码的校验矩阵 H,将 7 个非零 3 维列向量 “从小到大排列”。这是工程上的 “标准选择”,而非 “唯一选择”。 ^yinZAT2w
校验矩阵的 “核心约束”:这 3 点才是 “基本确定” 的 ^zr3GmOjr
列向量:必须包含 “所有非零 r 维向量”(r 是校验位位数) ^ZcM63lIg
结构:标准形式下需包含 “单位矩阵 Iᵣ”(后 r 列) ^Ap1Th81T
秩:必须是 “满行秩”(秩 = r=n-k) ^Qd49acy2
这个 H 矩阵的列向量包含了所有 7 个非零 3 维向量,因此它是合法的(7,4)汉明码 H 矩阵(能纠正 1 位错误),但因排列顺序不标准,工程上几乎不会使用—— 它更像 “理论上的合法矩阵”,而非实际应用中的选择 ^m9JPBSXb
模2运算:除以2的余数,千万别忘了 ^AXySRAm2
错误检验: ^GfrE1Vwt
第四步:定位错误 ^xsuN9doM
第五步:纠正错误,恢复发送码 ^tlCX8Zgh
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