Introduction to Abstract Algebra, Solutions Manual, 4th Edition

2.11 An Application to Binary Linear Codes

1. (a) 5 (c) 6

2. (a) 3 (c) 7

3. By (1) of Theorem 1,

5. a. It suffices to look at individual digits:

6. The table lists the codewords across the top and the distances of from them.

a. The unique nearest neighbor of 0110101 is 0100101 (so it corrects the single error).

c. 1011001 has both 1010101 and 1011010 at distance 2, so the 2 errors are detected but not corrected.

7. a. The minimum weight of C is 4. Detects 3 errors, corrects 1 error.

9.

a. We can have k = 4, n = 7 for the code, and (Example 9), the minimum weight for the code is 3. Thus it corrects t = 1 errors. Hence shows the code is perfect.

c. If the minimum distance is 5 = 2 · 2 + 1, it corrects t = 2 errors. We have while 2⁷⁻² = 2⁵ = 32. So no such code exists.

10. a. If k = 2, t = 1, the Hamming bound is , that is 1 + n ≤ 2ⁿ⁻². The first n ≥ 1 for which this holds ...