The Gilbert Bound

Abstract

Before we actually construct and analyze several specific codes in detail, we will demonstrate the existence of “good” linear codes by a simple argument first introduced by E.N. Gilbert. Suppose we have a linear code with 2i codewords of block length n and a minimum nonzero weight d which contains a coset of weight w ≥d. Then we may form an augmented code, whose 2i+1 codewords are the union of the 2i original codewords and the 2i words in the coset of the original code. The new code is easily seen to be linear and its minimum nonzero weight is still d. The augmented code has 2n−i−1 cosets, and if any of these has minimum weight w≥d, then we may again augment the augmented code without decreasing d. In fact, we may augment the code again and again, each time doubling the number of its codewords, until we obtain a code whose minimum nonzero weight is greater than the weight of any of its cosets. Since each coset of the final code then has a leader of weight < d, it follows that the number of cosets cannot exceed the number of words of weight < d and KeywordsLinear CodeBlock CodeBlock LengthSpecific CodeOriginal CodeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.