7/28/2019 Gallian Ch 31

1/38

7/28/2019 Gallian Ch 31

2/38

A k-dimensional subspace Vof the vector space

...nF F F F (n copies) over a field F. Themembers ofVare called code words. When Fis

2Z ,

the code is called binary.

7/28/2019 Gallian Ch 31

3/38

Hamming distance

7/28/2019 Gallian Ch 31

4/38

The number of components in which two vectors of avector space differ. Notation: d(u,v).

7/28/2019 Gallian Ch 31

5/38

Hamming weight

7/28/2019 Gallian Ch 31

6/38

The number of non-zero components of a vectoru.Notation: wt(u).

7/28/2019 Gallian Ch 31

7/38

Hamming weight of a linear code

7/28/2019 Gallian Ch 31

8/38

The minimum weight of any nonzero vector in the code.

7/28/2019 Gallian Ch 31

9/38

Properties of Hamming Distance and Hamming Weight

7/28/2019 Gallian Ch 31

10/38

For any vectors u, v, and w, ( , ) ( , ) ( , )d u v d u w d w v

and ( , ) ( )d u v wt u v .

7/28/2019 Gallian Ch 31

11/38

Correcting Capabilities of a Linear Code

7/28/2019 Gallian Ch 31

12/38

If the Hamming weight of a linear code is at least 2 1t ,then the code can correct any tor fewer errors.

Alternatively, the code can detect any 2t or fewererrors.

7/28/2019 Gallian Ch 31

13/38

Standard generator matrix(or standard encoding matrix)

7/28/2019 Gallian Ch 31

14/38

A kby n matrix in which the first kcolumns are theidentity matrix. When using this transformation matrix,

the message constitutes the first kcomponents of thetransformed vectors.

7/28/2019 Gallian Ch 31

15/38

Systematic code

7/28/2019 Gallian Ch 31

16/38

An (n,k) linear code in which the kinformation digitsoccur at the beginning of each code word.

7/28/2019 Gallian Ch 31

17/38

Parity-check matrix

7/28/2019 Gallian Ch 31

18/38

Let [ | ]k

G I A be the standard generator matrix.

The ( )n n k matrix Hwhose first krows are A and

whose next n k rows are the identity matrix.

7/28/2019 Gallian Ch 31

19/38

Decoding procedure using parity-check matrix(steps 1-3 of 4)

7/28/2019 Gallian Ch 31

20/38

Step 1: For any received word w, compute wH.

Step 2: IfwHis the zero vector, assume no error wasmade.

Step 3: If there is exactly one instance of a nonzeroelement s F and a row IofHsuch that wHis s times

row i, assume the sent word was (0,..., ,...,0)w s ,

where s occurs in the i-th component. If there is morethan one such instance, do not decode.

7/28/2019 Gallian Ch 31

21/38

Decoding procedure using parity-check matrix(steps 3 and 4 of 4)

7/28/2019 Gallian Ch 31

22/38

Step 3: When the code is binary, category 3 reduces tothe following. IfwHis the i-th row ofHfor exactly one i,

assume that an error was made in the i-th component ofw. IfwHis more than one row ofH, do not decode.

Step 4: IfwHdoes not fit into either category 2 orcategory 3, we know that at least two errors occurred intransmission and we do not decode.

7/28/2019 Gallian Ch 31

23/38

Orthogonality Relation

7/28/2019 Gallian Ch 31

24/38

Let Cbe an (n,k) linear code overFwith generatormatrix G and parity-check matrix H. Then, for any

vectorvin nF , we have 0vH (the zero vector) if anyonly ifvbelongs to C.

7/28/2019 Gallian Ch 31

25/38

Parity-Check Matrix Decoding

7/28/2019 Gallian Ch 31

26/38

Parity-check matrix decoding will correct any singleerror if and only if the rows of the parity-check matrix

are nonzero and no one row is a scalar multiple of anyother.

7/28/2019 Gallian Ch 31

27/38

Standard array

7/28/2019 Gallian Ch 31

28/38

A table constructed as follows.

The first row is the set ofCcode words, beginning incolumn 1 with the identity 0...0 .

To form additional rows, choose an element vofVnotlisted so far. Among all the elements of the coset v C ,choose one of minimum weight, say v. Complete thenext row of the table by placing under the columnheaded by the code word cthe vector 'v c .

Continue this process until all vectors in Vhave beenlisted.

7/28/2019 Gallian Ch 31

29/38

Coset leader

7/28/2019 Gallian Ch 31

30/38

A word in the first column of the standard array.

7/28/2019 Gallian Ch 31

31/38

Coset Decoding Is Nearest-Neighbor Decoding

7/28/2019 Gallian Ch 31

32/38

In coset decoding, a received word wis decoded as a

code word csuch that ( , )d w c is a minimum.

7/28/2019 Gallian Ch 31

33/38

Syndrome ofu

7/28/2019 Gallian Ch 31

34/38

An (n,k) linear code overFhas parity matrix H.

For any vectorn

u F , the vectoruH.

7/28/2019 Gallian Ch 31

35/38

Same Coset---Same Syndrome

7/28/2019 Gallian Ch 31

36/38

Let Cbe an (n,k) linear code overFwith parity-checkmatrix H. Then, two vectors of nF are in the same

coset ofCif and only if they have the same syndrome.

7/28/2019 Gallian Ch 31

37/38

Procedure for decoding a word in coset decoding

7/28/2019 Gallian Ch 31

38/38

1. Calculate wH, the syndrome ofw.2. Find the coset leadervsuch that wH vH .

3. Assume that the vector sent was w v .