An Introduction to Abstract Algebra via Applications by David R. Finston and Patrick J. Morandi

By David R. Finston and Patrick J. Morandi

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ERROR CORRECTING CODES We can write out all solutions to this system of equations, since each of x3 and x5 can take on the two values 0 and 1. This gives us four solutions, which we write as row vectors: (x1 ; x2 ; x3 ; x4 ; x5 ) = (1 + x3 ; 1 + x3 + x5 ; x3 ; 1; x5 ); where x3 = 0; 1 and x5 = 0; 1. Note that (1 + x3 ; 1 + x3 + x5 ; x3 ; 1; x5 ) = (1; 1; 0; 1; 0) + x3 (1; 1; 1; 0; 0) + x5 (0; 1; 0; 0; 1): Since (1; 1; 0; 1; 0) corresponds to the values x3 = x5 = 0 in *, this yields a particular solution to the linear system.

Thus, the associative property is not something that holds in every reasonable example. Merely having an operation on a set is not, in general, a very useful thing. The operation needs to satisfy some appropriate properties in order to be useful. We know various examples of useful properties, such as the commutative and associative laws. Now that we have the notion of a binary operation on a set, we can give the de…nition of a ring. This is the structure that generalizes the examples mentioned above.

1)=2c. Then C is a t-error Proof. Let w be a word, and suppose that v is a codeword with D(v; w) t. We need to prove that v is the unique closest codeword to w. We do this by proving that D(u; w) > t for any codeword u 6= v. If not, suppose that u is a codeword with u 6= v and D(u; w) t. Then, by the triangle inequality, D(u; v) D(u; w) + D(w; v) t + t = 2t < d: This is a contradiction to the de…nition of d. Thus, v is indeed the unique closest codeword to w. u1 t w u2 To …nish the proof, we need to prove that C does not correct t + 1 errors.

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