Abstract Algebra Theory and Applications - Computer Science ...

7.2 LINEAR CODES 119Proof. Observe thatd min = min{d(x, y) : x ≠ y}= min{d(x, y) : x + y ≠ 0}= min{w(x + y) : x + y ≠ 0}= min{w(z) : z ≠ 0}.□Linear CodesFrom Example 8, it is now easy to check that the minimum nonzero weightis 3; hence, the code does indeed detect and correct all single errors. Wehave now reduced the problem of finding “good” codes to that of generatinggroup codes. One easy way to generate group codes is to employ a bit ofmatrix theory.Define the inner product of two binary n-tuples to bex · y = x 1 y 1 + · · · + x n y n ,where x = (x 1 , x 2 , . . . , x n ) t and y = (y 1 , y 2 , . . . , y n ) t are column vectors. 3For example, if x = (011001) t and y = (110101) t , then x · y = 0. We canalso look at an inner product as the product of a row matrix with a columnmatrix; that is,x · y = x t y⎛= ( )x 1 x 2 · · · x n ⎜⎝= x 1 y 1 + x 2 y 2 + · · · + x n y n .⎞y 1y 2⎟. ⎠y nExample 9. Suppose that the words to be encoded consist of all binary3-tuples and that our encoding scheme is even-parity. To encode an arbitrary3-tuple, we add a fourth bit to obtain an even number of 1’s. Notice thatan arbitrary n-tuple x = (x 1 , x 2 , . . . , x n ) t has an even number of 1’s exactly3 Since we will be working with matrices, we will write binary n-tuples as column vectorsfor the remainder of this chapter.