Related papers: Equivalence for flag codes
Subspace codes have important applications in random network coding. It is interesting to construct subspace codes with both sizes, and the minimum distances are as large as possible. In particular, cyclic constant dimension subspaces codes…
An equivalence relation called isometry is introduced to classify constacyclic codes over a finite field; the polynomial generators of constacyclic codes of length $\ell^tp^s$ are characterized, where $p$ is the characteristic of the finite…
Any subset Q' of the domain Q of a learning space defines a projection of that learning space on Q' which is itself a learning space consistent with the original one. Moreover, such a construction defines a partition of Q having each of its…
The theory of q-analogs develops many combinatorial formulas for finite vector spaces over a finite field with q elements--all in analogy with formulas for finite sets (which are the special case of q=1). A direct-sum decomposition of a…
A permutation-invariant quantum code on $N$ qudits is any subspace stabilized by the matrix representation of the symmetric group $S_N$ as permutation matrices that permute the underlying $N$ subsystems. When each subsystem is a complex…
In this paper, we investigate the existence of self-dual MRD codes $C\subset L^n$, where $L/F$ is an arbitrary field extension of degree $m\geq n$. We then apply our results to the case of finite fields, and prove that if $m=n$ and…
A structure $\mathcal{A}=\left(A;E_i\right)_{i\in n}$ where each $E_i$ is an equivalence relation on $A$ is called an $n$-grid if any two equivalence classes coming from distinct $E_i$'s intersect in a finite set. A function $\chi: A \to n$…
Quantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set. It has been shown that this is possible for almost all choices of base sets and furthermore that…
Program equivalence in linear contexts, where programs are used or executed exactly once, is an important issue in programming languages. However, existing techniques like those based on bisimulations and logical relations only target at…
The primary goal of this paper is to abstract notions, results and constructions from the theory of categories to the broader setting of plots. Loosely speaking, a plot can be thought of as a non-associative non-unital category with a…
We will study convolutional codes with the help of state space realizations. It will be shown that two such minimal realizations belong to the same code if and only if they are equivalent under the full state feedback group. This result…
In this paper, we propose a general construction of linear perfect codes over infinite skew fields and quasi skew fields with right (left) unity. A complete classification of such codes over associative skew fields is given. Since the…
In this paper, we show that LCD codes are not equivalent to linear codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD$(n,2)$ over $\mathbb{F}_3$ and $\mathbb{F}_4$. We…
We completely characterize possible indices of quasi-cyclic subcodes in a cyclic code for a very broad class of cyclic codes. We present enumeration results for quasi-cyclic subcodes of a fixed index and show that the problem of enumeration…
Flash memory is a non-volatile computer memory comprising blocks of cells, wherein each cell can take on q different values or levels. While increasing the cell level is easy, reducing the level of a cell can be accomplished only by erasing…
We investigate group coding for arbitrary finite groups acting linearly on a vector space. These yield robust codes based on real or complex matrix groups. We give necessary and sufficient conditions for correct subgroup decoding using…
It is well known that two finite sequences of vectors in inner product spaces are unitarily equivalent if and only if their respective inner products (Gram matrices) are equal. Here we present a corresponding result for the projective…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
Quantum synchronizable error-correcting codes are special quantum error-correcting codes that are designed to correct both the effect of quantum noise on qubits and misalignment in block synchronization. It is known that in principle such a…
We determine the minimum possible column multiplicity of even, doubly-, and triply-even codes given their length. This refines a classification result for the possible lengths of $q^r$-divisible codes over $\mathbb{F}_q$. We also give a few…