Hints for question 3.8 in Tannenbaum

The question says:

A block of bits with n rows and k columns uses horizontal and vertical parity bits for error detection. Suppose that exactly 4 bits are inverted due to transmission errors. Derive an expression for the probability that the error will be undetected.

Step 1. Find the probability that exactly 4 bits are in error. Assume that there are a total of nxk bits.

Step 2. How many different ways are there to have exactly 4 bits in error?

For example, if there are nk places to make an error with the first bit, then there must be nk-1 places to make an error with the second bit. In fact there are

nk(nk-1)(nk-2)(nk-3) possible combinations for placing 4 errors in an nk block of bits.

When does an undetected error occur in an nxk block code with horizontal and vertical parity bits?

When there are an even number of errors in a row AND and even number of errors in a column. Thus the pattern is like this:

bbbeebbbp
bbbeebbbp
ppppppppp

Where b is a bit, e is an error and p is a parity bit. What is the total number of rectangles that is like this?

The probability of an undetected error is the number of such rectangles divided by the number of ways to distribute the four bits....Now you have enough information to write a formula!