By David R. Finston and Patrick J. Morandi

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2. Construct a linear code of length 5 with more than 2 codewords that corrects one error. Can you construct a linear code of length 4 with more than 2 words that corrects one error? 3. Let C be the code consisting of the solutions 0 1 0 1 @ A= 0 1 1 1 1 1 to the matrix equation Ax = 0, where 1 1 1 0 1 0 1 A: 0 0 0 Determine the codewords of C, and determine the distance and error correction capability of C. 4. Let A be a matrix, and let C be the code consisting of all solution to Ax = 0. If A has neither a column of zeros nor two equal columns, prove that the distance of C is at least 3.

Give an example of a ring R and an element a 2 R that is neither a zero divisor nor a unit. 14. Let R be a ring in which 0 = 1. Prove that R = f0g. 15. Prove that the multiplicative identity of a ring is unique. 16. If a and b are elements of a ring R, prove that ( a) ( b) = ab. 17. Prove that if a and b are elements of a ring, then (a + b)2 = a2 + ab + ba + b2 . 18. Let R be a ring in which a2 = a for all a 2 R. Prove that is commutative. 1 de…ne notation in section a = a. Then prove that R 54 CHAPTER 3.

5. Let R and S be rings. 12, prove that R distributive properties. 6. If R is a ring, prove that a( b) = ( a)b = 7. If R is a ring, prove that a(b c) = ab S satis…es the (ab) for all a; b 2 R. ac for all a; b; c 2 R. 8. 11 with the operations A + B = (A B) [ (B A B = A \ B: A) = (A [ B) (A \ B); is a ring. 9. A unit of a ring R is an element a such that the multiplicative inverse a Prove that if a and b are units, then ab is a unit. 1 of a exists. ) 10. Prove that a 2 Zn is a zero divisor if and only if gcd(a; n) > 1.