Abstract Algebra Theory and Applications - Computer Science ...

120 CHAPTER 7 ALGEBRAIC CODING THEORYwhen x 1 + x 2 + · · · + x n = 0; hence, a 4-tuple x = (x 1 , x 2 , x 3 , x 4 ) t has aneven number of 1’s if x 1 + x 2 + x 3 + x 4 = 0, or⎛ ⎞1x · 1 = x t 1 = ( )x 1 x 2 x 3 x 4⎜ 1⎟⎝ 1 ⎠ = 0.1This example leads us to hope that there is a connection between matricesand coding theory.Let M m×n (Z 2 ) denote the set of all m×n matrices with entries in Z 2 . Wedo matrix operations as usual except that all our addition and multiplicationoperations occur in Z 2 . Define the null space of a matrix H ∈ M m×n (Z 2 )to be the set of all binary n-tuples x such that Hx = 0. We denote the nullspace of a matrix H by Null(H).Example 10. Suppose that⎛H = ⎝0 1 0 1 01 1 1 1 00 0 1 1 1For a 5-tuple x = (x 1 , x 2 , x 3 , x 4 , x 5 ) t to be in the null space of H, Hx = 0.Equivalently, the following system of equations must be satisfied:⎞⎠ .x 2 + x 4 = 0x 1 + x 2 + x 3 + x 4 = 0x 3 + x 4 + x 5 = 0.The set of binary 5-tuples satisfying these equations is(00000) (11110) (10101) (01011).This code is easily determined to be a group code.Theorem 7.6 Let H be in M m×n (Z 2 ).group code.Then the null space of H is aProof. Since each element of Z n 2 is its own inverse, the only thing thatreally needs to be checked here is closure. Let x, y ∈ Null(H) for somematrix H in M m×n (Z 2 ). Then Hx = 0 and Hy = 0. SoH(x + y) = H(x + y) = Hx + Hy = 0 + 0 = 0.