t sets to output from the channel coder? Linear block coders are a group of block coders that follow a special set of rules when choosing which set of outputs to use. The rules are as follows, using a (6,3) code for illustrative purposes: Vn = the set of all possible 64 6-bit sequences U = the set of eight 6-bit sequences output at the channel coder Using this notation, the rule is this: U must be a subspace of V This means two very simple things: 1. U must contain {000000} 2. Adding (modulo 2) any two elements in U must create another element in U. Of course, examples make this much clearer to understand. Look at Figure 6.8(b)-is this a linear block code? Chapter Six 000111 6 bit output for input 000 001110 010101 Channel 011100 coder 100011 101010 110001 111000 possible 3-bit 6-bit outputs for inputs given 3-bit input 00O000 6 bit output for input 000 110001 011010 Channel 101011 coder 101100 011101 110110 000111 Figure 6.8 Which channel coder do you choose? First, looking at the channel coder outputs, we see that the element 000000 is in the output of the channel coder (the set U). That satisfies the first part of our rule. Now, let's try adding any two elements in U and see if we get another element in U. The addition, by the way, is modulo 2. Here, 110110 (7th element) + 011010 (3rd element) 101100 (5th element). Yes, we decide, it definitely is a block code. Example 6.3 Determine whether or not the input bit-output bits in Table E6.1 could represent a linear block coder. Solution: For a linear block code, you must make sure 0000 is in your output, and that the addition of any two output elements (modulo 2) leads to another output element. We immediately see 0000 as an output element, SO we know that we're OK with the first requirement. Next, we've got to make sure adding any two elements (modulo 2) leads to another element. Trying this out, we find, for example, 0101 Channel Coding and Decoding: Part 1-Block Coding and Decoding (element: 2) 1010 (element 3) 1111 (element input bits output bits 4); and 0000 