Abstract Algebra Theory and Applications - Computer Science ...

128 CHAPTER 7 ALGEBRAIC CODING THEORYSo we can add as many as four columns and still maintain a minimumdistance of 3.In general, if H is an m × n canonical parity-check matrix, then thereare n − m information positions in each codeword. Each column has mbits, so there are 2 m possible distinct columns. It is necessary that thecolumns 0, e 1 , . . . , e n be excluded, leaving 2 m − (1 + n) remaining columnsfor information if we are still to maintain the ability not only to detect butalso to correct single errors.7.4 Efficient DecodingWe are now at the stage where we are able to generate linear codes thatdetect and correct errors fairly easily, but it is still a time-consuming processto decode a received n-tuple and determine which is the closest codeword,because the received n-tuple must be compared to each possible codewordto determine the proper decoding. This can be a serious impediment if thecode is very large.Example 16. Given the binary matrix⎛1 1 1 0⎞0H = ⎝ 0 1 0 1 0 ⎠1 0 0 0 1and the 5-tuples x = (11011) t and y = (01011) t , we can compute⎛ ⎞0Hx = ⎝ 0 ⎠0and⎛Hy = ⎝Hence, x is a codeword and y is not, since x is in the null space and y isnot. Notice that Hx is identical to the first column of H. In fact, this iswhere the error occurred. If we flip the first bit in y from 0 to 1, then weobtain x.If H is an m × n matrix and x ∈ Z n 2 , then we say that the syndrome ofx is Hx. The following proposition allows the quick detection and correctionof errors.101⎞⎠ .