11.1 Groups and coding

Groups and Coding (group and coding)

Coding Theory: to help detect and correct errors by introducing redundancy information (parity bit)

• Basic definitions:

• The group defined by the group B defined in mod2 additions \ (B ^ m \) :

In the signal transmission channel (Transmission channel), the noise may be generated (Noise), such that the received code word error

Encoding function (Encoding function)

• Select a bijective (one-to-one) function E: \ (B ^ m -> n-B ^ (n-> m) \) :
  • So called e (m, n) encoding function, the \ (B ^ m \) each code word represents the \ (B \ ^ n) ; and
    
  • If \ (b ∈ B ^ m \) , then E (b) is referred to as a codeword b represents (code word)
• Obviously, if there is no noise during transmission, then the transmission side encoding for any e = X (B) ∈ \ (^ n-B \) , the receiver receives the \ (x_t = X \) ; and because the encoding function e is known, it can be directly decoded information b
  
• However, an error occurs during transmission is common, if the code words before and after the transmission of x \ (x_t \) there are different positions k, then we say that x = e (b) is a transmission error k
  

Error detection (Error detect)

Suppose E: \ (B ^ m -> n-B ^ \) is a (m, n) coding function:

• Obviously, if x = e (b) transmission error during transmission is k, then the receiver receives \ (x_t \) is not a code word ( \ (x_t! = X \) , can not be decoded directly b)
• For \ (x ∈ B the n-^ \) , we call x the number of 1s x weight (weight), and recorded as: \ (| x | \)

Parity (parity check code) and other functions encoded

Encoding function defined parity (even parity) E: \ (^ m B -> B. 1} ^ {m + \) , and ( \ (m + B_. 1} = {| B | ⊕0 \) ):

Consider a (m, 3m) encoding function e:

Given all the code words of the 3-bit information encoding:

Obviously, if the receiver receives a code word 01111101, this is not a code word (code word), described transmission error occurred during transmission

Hamming distance (Hamming distance)

• Set x, y belongs are \ (B ^ m \) codeword, we define the x and y Hamming distance (the Hamming Distance) \ ([theta] (x, y) = | x⊕y | \) , i.e., x ⊕y weight (weight)
  
• Hamming distance visual representation of x and y is: the number of bit positions different code words corresponding to x and y, also known as a distance (Distance)

Some properties of the Hamming distance:

note: (d) demonstrate the use of the associative operation ⊕

The minimum distance (Minimum distance)

Encoding function E: \ (B ^ m -> n-B ^ \) minimum distance refers to the minimum distance of any two code words (code word), i.e.

min { \ (θ (e (x) and (y)) | x, y ∈ B * m \) }

Theorem: (m, n) coding function capable of detecting or fewer errors k \ (<==> \) the minimum distance of the coding function \ (> = \) k +. 1

Encoding group (Group Codes)

If the (m, n) encoding function E: \ (B ^ m -> B ^ n-\) satisfies: \ (E (B ^ m) = \ {E (B) | E (B) ∈ B ^ n-\} \) , and there, \ (e (m ^ B) \) is \ (B ^ n \) subgroup, then we call code e is a group (group codes)

exp：

note: obtaining = N \ (E (B ^ m) \) , and then determines whether \ (B ^ n \) subgroup (about ⊕ is closed, the unit element, inverse element)

Theorem: Suppose e: \ (B ^ m -> n-B ^ \) is a group code , then: the minimum distance e is non-zero minimum weight code words for all codewords
prove definitely greater than the minimum distance :( 0 -> because there can be no two codewords after encoding is the same)


exp:

note: the benefits of a visible group code

Group coding (function) of the structure

Boolean matrix

• Matrix defined exclusive-OR operation (mod 2 addition): D ⊕ E
• Defined Boolean matrix multiplication (mod 2 multiplication Boolean): D * E
  then can have the following theorem: ⊕ and * for the matrix having the distributive property, i.e., (D ⊕ E) * F = (D * F) ⊕ (E * F) .

We define \ (x = x_1x_2 ... x_n ∈ B_n \) is a 1 × n matrix \ ([x_1 x_n x_2 ...] \) , we get the following theorems :

If \ (0 <m <n-, R & lt nm = \) , and let H be a Boolean n × r matrix, then there is:

• \ (f_H: B ^ n - > B ^ r, and f_H (X) * X = H, n-x∈B ^ \) , then the \ (f_H \) from the group \ (B ^ n \) to the group \ (B ^ r \) is a homomorphism
  proof (* ⊕ and distributive law):
  

Corollary 1:

Set \ (m, n, r, H and inference f_H just defined \) , then we can get the following information:

• \ (N = \ {x ∈ B ^ n | x * H = 0 \} is a normal subgroup of B ^ n \)

Proof:
\ (N is a homomorphism f_H core (kernel), so that N is a normal subgroup of B ^ n \)

Consistency test matrix (Parity check matrix)

Definitions: \ (Order m <n, r = nm, then the following Boolean n × r matrix is called \) consistency test matrix :

• Defined encoding function \ (e_H: B ^ m -> n-B ^ \) :
  \ (B_. 1 × m} = {... b_1b_2 B_M \)
  \ (then the code word x_ {1 × n} = e_H (b) ... ... x_r b_mx_1x_2 b_1b_2 = B_ = {m}. 1 × ×. 1 R & lt X_ {} \) , wherein:
  

• Complete matrix forms of expression:
  

Last Theorem:

1. Sufficiency:
   
   

1. necessity:
   

Corollary 2 (that is, its proof):

EXP:

Note: See, if you have a consistent test matrix, we can take the right by \ ([I_ {m × m } H_ {m × r}] \) to achieve \ (B ^ m -> B ^ n \) of group coding, but also through the right by \ (\ begin {bmatrix} H_ {m × r} \\ I_ {r × R} \ end {bmatrix} \) has occurred results to determine whether the transmission process 0 error