Abstract Algebra Theory and Applications - Computer Science ...

7.3 PARITY-CHECK AND GENERATOR MATRICES 127then the null space of H 1 is a single error-detecting code and the null spaceof H 2 is not.We can even do better than Theorem 7.12. This theorem gives us conditionson a matrix H that tell us when the minimum weight of the codeformed by the null space of H is 2. We can also determine when the minimumdistance of a linear code is 3 by examining the corresponding matrix.Example 15. If we let⎛H = ⎝1 1 1 01 0 0 11 1 0 0and want to determine whether or not H is the canonical parity-check matrixfor an error-correcting code, it is necessary to make certain that Null(H)does not contain any 4-tuples of weight 2. That is, (1100), (1010), (1001),(0110), (0101), and (0011) must not be in Null(H). The next theoremstates that we can indeed determine that the code generated by H is errorcorrectingby examining the columns of H. Notice in this example that notonly does H have no zero columns, but also that no two columns are thesame.Theorem 7.13 Let H be a binary matrix. The null space of H is a singleerror-correcting code if and only if H does not contain any zero columns andno two columns of H are identical.Proof. The n-tuple e i + e j has 1’s in the ith and jth entries and 0’selsewhere, and w(e i + e j ) = 2 for i ≠ j. Since⎞⎠0 = H(e i + e j ) = He i + He jcan only occur if the ith and jth columns are identical, the null space of His a single error-correcting code.□Suppose now that we have a canonical parity-check matrix H with threerows. Then we might ask how many more columns we can add to thematrix and still have a null space that is a single error-detecting and singleerror-correcting code. Since each column has three entries, there are 2 3 = 8possible distinct columns. We cannot add the columns⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞0 1 0 0⎝ 0 ⎠ , ⎝ 0 ⎠ , ⎝ 1 ⎠ , ⎝ 0 ⎠ .0 0 0 1