Lecturer

DIGITAL COMMUNICATIONSLinear Block Codes

Assist.Prof.Dr. Nuray At

EEM 467

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Designing codes for the reliable transmission of digital information over a noisy channel.

Codes can either correct or merely detect errors

Codes that can detect errors are called error-detecting codes Codes that can correct errors are called error-correcting codes

Error correction is more complex than error detection!

Error control codes are classified into Block Codes Convolutional Codes

The channel is discrete when the alphabets of X and Y are both finite The channel is memoryless when the current output depends only on the

current input and not on any of the previous inputs is a channel transition probability

Error Control Coding

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The channel encoder introduces systematic redundancy into the data stream The combined objective of the channel encoder and decoder is to minimize

the effect of channel noise

Channel Coding Theorem:Given a DMS X with entropy H(X) and a DMC with capacity C, if , there exists a coding scheme for which the source output can be transmitted over the channel with an arbitrary small probability of error.

Channel Coding

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Data sequence is divided into sequential blocks each k bits long Each k-bit block is converted into an n-bit block, where n > k

The resultant block code is called (n,k) block code and the ratio k/n is called code rate.

Block Codes

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Binary Field: The set K = {0, 1} is a binary field. The binary field has two operations, addition and multiplication

Addition

Multiplication

Linear Block Codes

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Linear Codes:Let and be two codewords in C.

A code C is called linear if the sum of two codewords is also a codeword in C. A linear code C must contain the zero codeword

Hamming Weight and Distance:Let a, b, and c be codewords of length n. The Hamming weight of c, denoted by w(c), is the number of 1's in c. The Hamming distance between a and b, denoted by d(a, b), is the number of

positions in which a and b differ.

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Thus, the Hamming weight of a codeword c is the Hamming distance between c and 0, that is

Similarly, the Hamming distance can be written in terms of Hamming weight as

Minimum Distance:The minimum distance dmin of a linear code C is defined as the smallest Hamming distance between any pair of codewords in C.

Theorem:The minimum distance dmin of a linear code C is the smallest Hamming weight of the nonzero codeword in the C.

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Error Detection and Correction Capabilities:The minimum distance dmin of a linear code C determines the error detection and correction capabilities of C. A linear code C of minimum distance dmin can detect up to t errors iff

A linear code C of minimum distance dmin can correct up to t errors iff

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Generator Matrix: In an (n,k) linear block code C,

If the data bits appear in specified location of c, the code C is called systematic. That is, Here we assume that the first k bits of c are the data bits.

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In a matrix form

Hence, and

The k x n matrix G is called the generator matrix.

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Parity-Check Matrix:Let H denote an m x n matrix defined by

where . The matrix H is called the parity-check matrix of C.

We have

Thus,

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Let r denote the received word of length n when codeword c of length n was sent over a noisy channel.

where e is called the error pattern. Consider first the case of a single error in the ith position. Then,

Evaluate as

where s is called syndrome of r.

Using s and noting that is the ith row of HT, we can identify the error position by comparing s to the rows of HT.

Note that the zero syndrome indicates that r is a codeword and is presumably correct.

Syndrome Decoding

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Example: Consider a linear block code with the following parity-check matrix

a. Determine the generator matrix G.b. Suppose that the received word is r = [1 1 0 1 1 0]. Decode this received

word, i.e., find c and d.

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Code length: Number of parity symbols: n – k = m Error correcting capability: t = 1

The parity-check matrices for binary Hamming codes are quite easy to construct. For a Hamming code of length construct a matrix whose columns consist of all nonzero m-tuples. For example, a parity-check matrix for a (15,11) Hamming code

The ordering of columns is arbitrary; another arrangement would still define a (15,11) Hamming code.

The Hamming Codes