The binary coding theory and subspace codes for random network coding exhibit similar structures. The method used to obtain a Singleton bound for subspace codes mimic the technique used in obtaining the Singleton bound for binary codes. This motivates the question of whether there is an abstract framework that captures these similarities. As a first step towards answering this question, we use ...

In the context of constant-dimension subspace codes, an important problem is to determine largest possible size $ A_q(n, d; k) codes whose codewords are k $-subspaces {\mathbb F}_q^n with minimum distance d $. Here in order obtain improved constructions, we investigate several approaches combine codes. This allow us present improvements on lower bounds for many parameters, including A_q(10, 4; ...

We prove that on any two-dimensional lattice of qudits a prime dimension, every translation invariant Pauli stabilizer group with local generators and the code distance being linear system size is decomposed by Clifford circuit constant depth into finite number copies toric (Abelian discrete gauge theory). This means under circuits, complete topological codes. Previously, same conclusion was ob...

A quasi-Gray code of dimension n and length l over an alphabet Σ is a sequence of distinct words w1, w2, . . . , wl from Σ n such that any two consecutive words differ in at most c coordinates, for some fixed constant c > 0. In this paper we are interested in the read and write complexity of quasi-Gray codes in the bit-probe model, where we measure the number of symbols read and written in orde...

A spread is a set of lines of PG(n, q), which partition the point set. A parallelism is a partition of the set of lines by spreads. Some constructions of constant dimension codes that contain lifted MRD codes are based on parallelisms of projective spaces. We construct all (321) parallelisms in PG(3, 5) with automorphisms of order 13 and find the order of their full automorphism groups.

Topological stabilizer codes with different spatial dimensions have complementary properties. Here I show that the spatial dimension can be switched using gaugefixing. Combining 2D and 3Dgauge color codes in a 3Dqubit lattice, fault-tolerant quantum computation can be achievedwith constant time overhead on the number of logical gates, up to efficient global classical computation, using only loc...