Concepts of hamming distance

Hamming Distance Problem: In general, it is assumed that it is more likely to have fewer errors than more errors. This “worst-case” approach to coding is intuitively appealing within itself. Nevertheless, it is closely connected to a simple probabilistic model where errors are introduced into the message independently for each symbol and with a fixed probability p < 1/2. In order to talk about the “number of errors” hamming distance is introduced.

Definition: The Hamming distance between two integers is the number of positions at which the corresponding bits are different. It is not dependent on the actual values of xi and yi but only if they are equal to each other or not equal.

Proposition: The function d is a metric. That is, for every x, y, z ∈ A_N:

• 0 ≤ d(x, y) ≤ N
• d(x, y) = 0 if and only if x = y
• d(x, y) = d(y, x)
• d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

Rule for Decoding: Below are the rules:

• Let C be a code of length N over an alphabet A.
• The nearest neighbor decoding rule states that every x ∈ A_N is decoded to cx ∈ C that is closest to x.
• That is, D(x) = cx where cx is such that d(x, cx) = minc ∈ C{d(x, c)}.
• If there exists more than one c at this minimal distance, then D returns ⊥.
• Code distance and error detection and correction.
• Intuitively, a code is better if all the codewords are far apart.

Formalized Notion:
Let C be a code. The distance of the code, denoted d(C), is defined by:
 d(C) = min{d(c1, c2) | c1, c2 ∈ C, c1 6= c2}

An (N, M)-code of distance d is called an (N, M, d)-code. The values N, M, d are called the parameters of the code.