Plot hole: Early on in the book the NSA is described as a top secret agency, an agency very few Americans know exist. However throughout the rest of the book there are many references to groups who openly protest against the NSA, bad widespread publicity, and the majority of Americans being against the agency. If hardly anybody knows about the NSA then how can there be so much open debate about it?
Continuity mistake: In Chapter 120, the main characters discover what they realize is an 8-by-8 Caesar box code. The first two times we see the letters of this code, the fourth sequence of four letters reads MFHA. The first letter, M, is a typo - it should actually be P. When the letters are grouped in an 8-by-8 grid in Chapter 121, this sequence has now changed to the correct PFHA. The correct letter P gives you the word "responsible"; the incorrect letter M would yield "resmonsible".
Factual error: The whole book is about a computer breaking a very strong code, and that for every code a large enough computer can be built to break it. That's simply not true: The "one time pad" cannot be broken by brute force, only by traditional stealing of the key.
Trivia: The cipher that Brown calls a "Caesar box cipher" is actually called a columnar transposition cipher. Julius Caesar did really invent ciphers, but the only one whose description has survived - and which to this day is called the "Caesar cipher" - is much simpler than the columnar transposition.