The (old) International Standard Book Number (ISBN) system, which was used to uniquely identify books up to 2007, is a code defined over the 11-letter alphabet F = {0, 1, ...,9, X}, where X represents the number 10. An ISBN codeword has length 10 and is written as aja2 ... 210. (In practice, these numbers are interspersed with several hyphens for readability.) The first nine digits are information digits which record data such as country, publisher, title and edition. The tenth digit is a check digit, chosen so that 10 j.a; = 0 (mod 11). j=1 For example, S. Lang's Algebra (Springer, 2002) has ISBN 0-387-95385-X. (a) Find the minimum distance for this code, and use this to show that the nearest codeword decoder for this code can detect 1 but not 2 errors. Moreover, show that it detects one transposition error (a;a;+1 becomes aj+1j, for some j, 15 j