A code is said to be locally testable if an algorithm can distinguish between a codeword and a vector being essentially far from the code using a number of queries that is independent of the code?s length. The question of characterizing codes that are locally testable is highly complex. In this work we provide a sufficient condition for linear codes to be locally testable. Our condition is based on the weight distribution (spectrum) of the code and of its dual.

Codes of (large) length n and minimum distance \frac{n}{2} - \Theta (\sqrt n ) have size which is at most polynomial in n. We call such codes almost-orthogonal. We use our condition to show that almost-orthogonal codes are locally testable, and, moreover, their dual codes can be spanned by words of constant weights (weight of a codeword refers to the number of its non-zero coordinates).