In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types. [1]

In the rst section we develop the basics of linear codes, in particular we introduce the crucial concept of the dual of a code. The second and third sections then discuss the general principles behind encoding and decoding linear codes. We encounter the important concept of a syndrome.

We represent the d(C?) by d? Let G be the generator matrix of the Simplex Code. We have t(G) = 3. Prove this.

We will begin by results on the existence and limitations of codes, both in the Hamming and Shannon approaches. This will highlight some criteria to judge when a code is good, and we will follow up with several explicit constructions of \good" codes (we will encounter basic nite eld algebra during these constructions).

Codes like this take exponential space to describe. In applications, we’d like explicit codes which have polynomial-time computable encoding. The most im-portant class of explicit codes are linear codes, for which efficient encoding is immediate. Definition 5.Let qbe a prime power and let Σ = F q be a finite field. Alinear code is a subspace ...

16.1 Linear coding De nition 16.1 (Linear code). Let X =Y =Fn k ∶ → ∀ ∈ q, M=q . Denote the codebook by C <{c u∶ = ∈ u Fk q. A code f Fk q F n q is a linear code if u Fkq, c u uG(row-vector convention), where G Fk n q. ∈ is a generator matrix ×} Proposition 16.1. c∈C ⇔c row span of G c ∈ KerH; for some H F(n−k)×n q s.t ...

We can use algebra to design linear codes and to construct efficient encoding and decoding algorithms. The absolute majority of codes designed by coding theorists are linear codes.

Coding theory is an important study which attempts to minimize data loss due to errors introduced in transmission from noise, interference or other forces. With a wide range of theoretical and practical applications from digital data transmission to modern medical research, coding theory has helped enable much of the growth in the 20th century.

A code is a linear code if it is determined by the null space of some matrix \(H \in {\mathbb M}_{m \times n}({\mathbb Z}_2)\text{.}\)

We will study a class of codes called linear block codes, because their structure o ers several advantages. Linearity will allow an easier analysis of the error correcting ability of a code. Furthermore, the use of matrices to encode/decode messages is much easier than a random codebook, and the code can be concisely described.