In coding theory, the problem of finding shortest linear codes for a fixed set parameters is central. Given dimension $k$, minimum weight $d$, and order $q$ finite field $\bF_q$ over which code defined, function $n_q(k, d)$ specifies smallest length $n$ an $[n, k, d]_q$ exists. The determining values this known as optimal codes. Using geometric methods through projective geometry, we determine ...

Motivated by an application to database linear querying, such as private information-retrieval protocols, we suggest a fundamental property of codes– the generalized covering radius. The covering-radius hierarchy code characterizes trade-off between storage amount, latency, and access complexity, in systems. Several equivalent definitions are provided, showing this combinatorial, geometric, alg...

We consider the Grassmann graph of k-dimensional subspaces an n-dimensional vector space over q-element field and its subgraph Γ(n,k)q which consists all non-degenerate linear [n,k]q codes. assume that 1<k<n−1. It is well-known every automorphism induced by a semilinear corresponding or isomorphism to dual second possibility realized only for n=2k. Our result following: if q≥3 k≠2, then can be ...