Linear Codes

Abstract

There is a lack of structure in the block codes we have considered in the first few chapters. Either we chose the code entirely at random, as in the proof of Theorem 1.12, or we built the code using the greedy algorithm, as in the proof of the Gilbert–Varshamov bound, Theorem 3.7. In this chapter, we introduce some algebraic structure to the block codes by restricting our attention to linear codes, codes whose codewords are the vectors of a subspace of a vector space over a finite field. Linear codes have the immediate advantage of being fast to encode. We shall also consider a decoding algorithm for this broad class of block codes. We shall prove the Griesmer bound, a bound which applies only to linear codes and show how certain linear codes can be used to construct combinatorial designs.