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编码理论笔记

日期:
分类: 备忘
标签: 编码理论 RemNote
Coding theory
  • Check ISBN
    • What is the check digit of ISBN 0-306-40615-??
    • Check the ISBN 9-219-53894-6.
    • Why we use mod 11 here?
      • The modulo-11 algorithm yields a near-100% guarantee that, if any two accounts differ in a single digit, their check digits will also be different. This gives us a minimum Hamming distance of , which allows us to detect any single-digit error.
    • Find two coding methodologies in daily life and describe each of them.
      • Morse code (trinary!),
      • RAID-5,
      • Barcode, etc.
  • Design codes
    • Nonsingular Codes
      • 每个源的元素都映射到不同的码字
        • 即每个源符号都有唯一的编码,不会有两个源符号映射到同一个码字
    • Uniquely Decodable Codes
      • 保证接收到的码字序列可以唯一解码为原始信息符号序列
    • Prefix Code
      • 没有任何码字是其他码字的前缀
      • ——这是防止编码产生歧义的关键特性,保证了即时解码的可能性
    • Design:
      • C1: singular
      • C2: non-singular
      • C3: UDC
      • C4: Prefix
      • +---------+----+-----+-----+-----+
        | Message | C1 | C2  | C3  | C4  |
        +---------+----+-----+-----+-----+
        |    A    |  0 |  0  |  10 |  0  |
        |    B    |  0 | 010 |  00 |  10 |
        |    C    |  0 |  01 |  11 | 110 |
        |    D    |  1 |  10 | 110 | 111 |
        +---------+----+-----+-----+-----+
    • Make a nonsingular code but not uniquely decodable. Then make a uniquely decodable code but not prefix code.
    • Examples
      • Note that Instantaneous ⇒ Uniquely Decodable ⇒ Non-singular
      • is an (and therefore uniquely decodable and non-singular).
      • is since no two codewords are the same. However it is not uniquely decodable since and are indistinguishable (it follows that it is not an instantaneous code either).
      • is since and have the same codeword (hence not uniquely decodable and not an instantaneous code).
      • is : when you get a it is the start of a new symbol and the previous symbol is given by counting how many 's since the previous . It is not a instantaneous code since the first codeword is a prefix of all the others.
      • is not an instantaneous code since is a prefix of . Since all codewords have even length we can rewrite it as a -ary code . It is easy to see that this is uniquely decodable since only ever appears as the second half of . Thus if an is followed by anything other than , it represents a .
  • Coding gain
    • jpg (1).jpg
    • The limit of coding gain of some code is its gain compared to Shannon limit.
    • What is the limit of coding gain of rate one half code at BER ?
  • Dual space
    • Dual space of , is an -dimensional subspace.
    • Each vector in is linearly independent and is in .
  • Linear block code
    • Generative matrix
      • 从 code words 中选 个向量,末尾正好为单位阵
    • Code word
    • Parity check matrix
      • 满足
    • Detect error
      • Syndrome of encoded message , .
        • Let . The syndrome of is
  • Hamming code
    • Construct (the parity-check matrix of) a hamming code of length 7? How about length 8?
      • Parity-check matrix 应满足 ,并且每列不重复、不为
      • Length 7
      • Length 8
    • What is the minimum for correcting 2 errors?
  • Global decoding
    • Suppose . Please give the encoded codeword .
    • 写出译码顺序
      • 根据已知 ,算
      • 总结:先解 ,后解
  • Tanner graph
    • 坦纳图表示的是 LDPC 的校验矩阵
    • tanner2.svg
    • Write the parity-check matrix according to the Tanner graph.
    • Find the shortest cycle in the Tanner graph.
  • Network coding
    • 利用两个数据包的异或(XOR)和作为冗余数据来加快传输
  • Max-flow min-cut theorem
    • (s-t cut):把节点分成两个集合 位于 中, 位于
    • :离开(流出) 的边(箭头)的权重之和
    • 最大流最小割(Max-flow min-cut)定理:最大流 的值 = 最小割的 capacity
      • 在网络中,源点 到汇点 的最大流量与网络中的最小割集所能承载的容量相等
    • 定义
      • 是一个有向图,其中 是源点, 是汇点;边的容量表示从源点到汇点的流量
      • 流量的值表示从源点 流向汇点 的总流量
      • s-t割 是顶点集合 的一个划分,使得 。割集 是从集合 中的一个顶点 指向集合 中的一个顶点 的所有边的集合,即
    • 例子
      • Not mininum
      • Minimum
        • image.png

2023 年 12 月 29 日写于 RemNote

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