Error is inevitable, protection is necessary

Extension of System Model

We will extend message bits to know errors possible even repair the errors

How do Errors Happen?

communication, data storage, computation

Detection & Correction

It is necessary to contain redundancy in message

In fact, if every illegal pattern is, in a sense to be described below, closer to one legal message than any other, the decoder could substitute the closest legal message, thereby repairing the damage.

Hammnig Distance

We need some technique for saying how similar two bit partterns are, so there is Hamming Distance

A more useful definition of the difference between teo bit patterns is the number of bits that are different between the two.
In the encoder, it is useful to calculate Hamming distance. But because of compression, Hamming distance must be set at least 3 to distingguish two legal code.

Single Bits

It is not important but commonly

能很好的避免偶然错误，但是出现系统错误（相邻的位同时变化），三重冗余将失效

Multiple Bits

To detect errors in a sequence of bits several techniques can be used. Some can perform error correction as well as detection.

Parity

用于出错概率不大

Rectangular Codes

二维阵列的奇偶校验位：
可以纠正一个错误，发现两个错误，无法处理三重错误

Hamming Codes

[[Hamming Codes]]

Block Codes

7 bit Hamming Codes: (7,4,3)block code