Abstract Algebra Theory and Applications - Computer Science ...

356 CHAPTER 20 FINITE FIELDSExample 7. In Example 6,x 7 − 1 = g(x)h(x) = (1 + x + x 3 )(1 + x + x 2 + x 4 ).Therefore, a parity-check matrix for this code is⎛0 0 1 0 1 1⎞1H = ⎝ 0 1 0 1 1 1 0 ⎠ .1 0 1 1 1 0 0To determine the error-detecting and error-correcting capabilities of acyclic code, we need to know something about determinants. If α 1 , . . . , α nare elements in a field F , then the n × n matrix⎛⎜⎝⎞1 1 · · · 1α 1 α 2 · · · α nα1 2 α2 2 · · · αn2 .. . ..⎟. ⎠α1 n−1 α2 n−1 · · · αnn−1is called the Vandermonde matrix. The determinant of this matrix iscalled the Vandermonde determinant. We will need the following lemmain our investigation of cyclic codes.Lemma 20.12 Let α 1 , . . . , α n be elements in a field F with n ≥ 2. Then⎛⎞1 1 · · · 1α 1 α 2 · · · α ndetα1 2 α2 2 · · · αn2 = ∏(α i − α j ).⎜⎝.. . ..⎟. ⎠ 1≤j