7. linear codes

7/31/2019 7. Linear Codes

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Binary Block Codes

Definition of Binary Block Code:A binary block code is a subset of for some n.Elements of the code are called code words

Definition of Linear Code:

A linear code is a linear subspace of

Definition of Minimum Distance:The minimum distance of a linear code is the minimum ofthe weights of the non-zero code words

Bn

Bn


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Example

The following two-dimensional codes in haveminimum distance 1

The code {000, 011, 101, 110} has minimum distance 2

3B


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Binary Block Codes

IfL is a k-dimensional subspace of , then we can find abasis for L consisting ofkcode words in

Every code word in L is a linear combination of the basiscode words

B

n

1 2, , ,

kb b b B

n


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Binary Block Codes

Definition of Generator Matrix:A generator matrix for a linear code is a binary matrixwhose rows are the code words belonging to some basisfor the code

A generator matrix for a k-dimensional linear code is ak x n matrix whose rank is k

A rankis the number oflinearly independent rows or

column in a matrix


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Example 1

The code {0000, 0001, 1000, 1001} is a two-dimensionallinear code in B4

The basis of this code is {0001, 1000}

To find the code words from the generator matrix


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Example 2

The code {0000, 0011, 0110, 1100, 0101, 1111, 1010,1001} is a three-dimensional linear code in B4

The basis of this code {0011, 0110, 1100}

Thus, the generator matrix is

The code words can be obtained:


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Example 2 (Contd)

If we use basis of {0101, 1001, 1010}, the generator matrixwill be

The code words are


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Example 3

The generator matrix ofrank3 is given below

The code words are


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Binary Block Codes

Definition of Elementary Row Operation:An elementary row operation on a binary matrix consistsof replacing a row of the matrix with the sum of that rowand any other row

If we have a generator matrix G for a linear code, allother generator matrices for L can be obtained byapplying a sequence of elementary row operations to G


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Binary Block Codes

Definition of Equivalent Codes:

Two codes are equivalent if each can be constructed from theother by reordering the bits of each code word in the sameway

Definition of Canonical Form:The generator matrix G of a k-dimensional linear code in Bn isin canonical form if it is of the form:

where I is a k x k identity matrix and A is an arbitrary

k x (n k) binary matrix

|G I A


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Example

We have

To make canonical form, we conduct the following steps

The code generated by the canonical form: {0000, 1000,0100, 1100}

It has minimum distance 1


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Binary Block Codes

Definition of Parity Check Matrix:The parity check matrix of a linear code with k x ngenerator matrix G is the k x n matrix H satisfying

where HT is the transpose ofH and 0 denotes thek x (n k) zero matrix

0

TGH


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Binary Block Codes

If G is canonical form, then

If G is not canonical, we can find H by:

reducing G to canonical from

finding the canonical form of H by using the equation above

reversing the column operations used to convert G tocanonical form to convert G the canonical form of H to the

parity check matrix of G

|T

H A I


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Example 1

The canonical matrix G is given

with

The H is obtained by transposing A and adjoining a 2 x 2identity matrix to get

By definition


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Example 2

The generator matrix for linear code {000, 101, 011, 110}is

The parity check matrix is

If we calculate all the elements of the B3, it results


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Binary Block Codes

IfL is a k-dimensional linear code inBn

, and G is agenerator matrix for L, every code word in L can beobtained by taking some and multiplying it by G,to get the code word bG.

If we now multiply this by the transpose of the paritycheck matrix H, we get

kb B

0 0T TbG H b GH b


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Example

The linear code {0000, 0011, 1100, 1111} has generatormatrix of

The parity check of this matrix is

We can check


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Encoding and Decoding

If we assume the corruption occured while transmissionthrough a BSC and we use ML, it follows that a corruptedstring should be restored to the code word which isclosest to it

Suppose that generator matrix G generates the linear

code L, and the code word w is corrupted by a noisevector e, the result is x = w + e

Ifx is not the code word, the decoder must find theclosest code word y (the error vector is smallest)

Ifx is not a code word, it belongs to the coset x + L,which is x = e + u where and e is the vector ofleast weight in the coset

Lu


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Encoding and Decoding

Definition of Syndrome:

If L is a linear code with generator matrix G and paritycheck matrix H, the syndrome ofx is given by

Procedure for removing noise: Draw up the syndrome table: list of the cosets, showing the

vector of minimum weight and the syndrome

When a string that is not a code word is received, compute itssyndrome

Add the vector of minimum weight from the correspondingcoset to the corrupted string to recover the uncorrupted codeword

Ts H x


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Example

Suppose that the parity check matrix of the linear code L

To draw up the syndrome table, we start with the

elements of L, which have syndrome 00

syndrome code words Error vector


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Example (Contd)

We choose a vector that does not belong to L andcompute its syndrome. If we choose 0001, whosesyndrome is 10, we add this vector to all the code words

If we choose 0110, the syndrome is 11


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Example (Contd)

Then, we choose 0100 whose syndrome is 01

If we receive the vector 1010, we compute the syndrome,which is 11

We add the vector of minimum weight in the coset withsyndrome 11 to 1010, so the corrected code word is1111

7. linear codes

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