Definition of a Code in Coding Theory

Given $n\in N$, a finite sequence $x_1,x_2,\dots ,x_i$ with every $x_i$ belonging to the alphabet $A$ is called a word (or vector or message) of length $n$ over $A$. Such a word can also be viewed as an element $(x_1,x_2,\dots ,x_i)\in A^n$. A $q$-ary code (over $A$) of length $n$ is just a subset $C\subset A^n$. A (valid) codeword is an element of $C$.