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 …

A linear code that meets the Singleton bound with equality is called maxi-mum distance separable or, for short, an MDS code. Every subset of n k + 1 coordinate positions supports a codeword …

Linear algebraically, it is simply the subspace spanned by the Hadamard code and the all 1's vector (i.e., the union of the Hadamard code H and its coset H + 1).

Let G be the generator matrix of the Simplex Code. We have t(G) = 3. Prove this.

The code is systematic: x1, x2, x3 can be freely chosen, so we can take: x1 = u1, x2 = u2, x3 = u3 whereas the parity bits x4, x5, x6 are uniquely determined by x1, x2, x3.

Jun 14, 2025 · In this comprehensive guide, we will explore the fundamentals, advanced topics, and real-world applications of Linear Codes. Linear Codes are a crucial concept in Coding …

consider the error detection problem for linear codes. Given a vector v ∈ Fq how can we recognize whether v is exactly a codeword? We can do this using the “dual” representation of …

Definition 1.1. A parity check is a h ∈ {0, 1}n such that h · c = 0 (mod 2) for all c ∈ C. The set of parity checks is denoted C⊥ := {h | h · c = 0∀c ∈ C}. C⊥ then h1 + h2 ∈ C⊥ so C⊥ is a linear …

A linear code is defined as a type of coding system characterized by its highly structured nature, which allows for the application of powerful mathematical tools, such as linear algebra, …

The most important class of codes is linear codes. Their ability to correct errors is no worse than that of general codes, but linear codes are easier to implement in practice and allow us to use …