 1 - O(n-3/4). binary vectors that represent the information that is to be CHAPTER 3.4 Channel protection fundamentals sent over a channel. The goal of using lattice codes for region, then the decoder will be able to successfully channel coding is to maximize the amount of information decode the bits sent by the encoder. that can be conveyed over the channel for a given power constraint. As an example, consider the problem of sending bits over an AWGN channel. One method of 3.4.3.2 Convolutional codes addressing this problem is to choose a lattice and then map vectors of bits to a finite subset of the lattice. The A class of codes that are often used with both Hamming lattice points will then represent the real values that are distance decoders and Euclidean distance decoders are sent over the channel and corrupted by noise. The convolutional codes. For simplicity, we restrict our decoder will receive a noisy sequence of points and at- description to binary convolutional codes. Like an (n,k) tempt to recover the bits by decoding the noisy values to linear block code, an (n, k) convolutional code maps the closest lattice points in Euclidean distance. The length-k blocks of information symbols into length-n decoding region for each lattice point is often referred to blocks of output symbols, but each output block depends as its Voronoi region and is defined to be the set of points on the current and previous information blocks. A whose Euclidean distance to the given lattice point is convolutional code can in general be defined by a linear closer than that to any other lattice point. finite state machine (LFSM). For a binary (n, k) convo- To illustrate the above concepts, consider the lutional code of memory v, the LFSM can be expressed as hexagonal lattice defined by the generator matrix V stages of k shift registers that are connected by n different modulo-2 adders, as in Figure 3.4-5. At each time instant, k bits are shifted in to the LFSM and n bits are output from the LFSM. The shift registers in com- bination