Outline:

- Linear code structure
- Coder and decoder
- Linear code performance

A (**binary**) code is linear if every **linear combination** of code words is also a code word

**modulo -2 addition**

** The linearity of a code depends only on the code words and not on how the information blocks are mapped to the code words**.

Example:

If ci=(11001) ∈ C cj=(11000) ∈ C →

(11001) ⊕ (11000) = (00001) ∈ C

Closure property (under the modulo 2 addition) and the presence of c0=(0000…0) → C is a vector subspace of the space formed by all the n-dimensional binary vectors.

The vector subspace C can be generated by a set of vectors, namely by a base

Example 1:

A (5,2) code defined by

C={00000, 10100, 01111, 11011} is linear because it verifies the closure property and the code word (00000) is in the code.

The code is **systematic** because the first k (k=2) symbols of the code word coincide with the symbols of the information sequence followed by some extra bits called parity check bits.

The matrix G in this case is written belowe.

It is possible to show that code words of C *are orthogonal to the code words of C ^{T}:*

* cH^{t}=0 *∀ c ∈ C

**cG**^{t}=0 ∀ c ∈ C^{T}

The matrix H is called the *parity check matrix* of the code C.

Since all rows of the generator matrix are code words we have

** **

**GH ^{t }= 0**

The Hamming codes are linear block codes with

C(2m-1, 2m-1-m) and dmin=3

built in the following way.

The columns of the parity check matrix H are all binary sequences of length m except the all zero sequence which number is equal to 2^{m}-1.

Show that the minimum Hamming distance of a linear block code is equal to the minimum number of columns of its parity check matrix that are linearly dependent.

The parity check matrix H can be expressed as:

*H=[h _{1}, h_{2}, ..., h_{n}] with hi a (n-k)-coumn vector*

Let **c** = [c_{1},.....c_{n}] a code word with l non-zero components.

Let us denote with c_{i1},…c_{i}l the l components equal to 1.

*Since cH ^{t} = 0 →*

*c _{1}h_{1}+c_{2}h_{2}+…-c_{n}h_{n} = 0 →*

*c _{i1}h_{i1}+c_{i2}h_{i2}+…+c_{in}h_{in} = 0*

If a code word has Hamming weight 1 *→* l column of H is linearly independent

If a code has the minimum distance d_{min} *→* there exists at least a code word with weight d_{min} *→*d_{min }is the minimum number of columns of H linearly dependent.

*5*. Digital Transmission over AWGN chanel

*6*. Evaluation of P(e) for optimum RX in AWGN

*7*. Error probability for M-PSK

*8*. Noncoherent ML Demodulation of FSK signals in AWGN

*9*. Transmission through bandlimited AWGN channels

*13*. Cyclic Codes

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