相关论文: Generalization the parameters of minimal linear co…

Let $(R,\m)$ be a formally unmixed local ring of positive prime characteristic and dimension $d$. We examine the implications of having small Hilbert-Kunz multiplicity (i.e., close to 1). In particular, we show that if $R$ is not regular,…

Lattices defined as modules over algebraic rings or orders have garnered interest recently, particularly in the fields of cryptography and coding theory. Whilst there exist many attempts to generalise the conditions for LLL reduction to…

Minimal codes are being intensively studied in last years. $[n,k]_{q}$-minimal linear codes are in bijection with strong blocking sets of size $n$ in $PG(k-1,q)$ and a lower bound for the size of strong blocking sets is given by…

In this paper, we defined the $Z_q$-linear codes and discussed its various parameters. We constructed $Z_q$-Simplex code and $Z_q$-MacDonald code and found its parameters. We have given a lower and an upper bounds of its covering radius for…

Finding the largest code with a given minimum distance is one of the most basic problems in coding theory. In this paper, we study the linear programming bound for codes in the Lee metric. We introduce refinements on the linear programming…

Minimal codes are a class of linear codes which gained interest in the last years, thanks to their connections to secret sharing schemes. In this paper we provide the weight distribution and the parameters of families of minimal codes…

Minimal linear codes are algebraic objects which gained interest in the last twenty years, due to their link with Massey's secret sharing schemes. In this context, Ashikhmin and Barg provided a useful and a quite easy to handle sufficient…

In this short survey we concern ourselves with minimal codes, a classical object in coding theory. We will explain the relation between minimal codes and various other mathematical domains, in particular with finite projective geometry.…

In the setting of minimal local grammar-based coding, the input string is represented as a grammar with the minimal output length defined via simple symbol-by-symbol encoding. This paper discusses four contributions to this field. First, we…

This paper investigates the existence of minimal $p$-encoders for convolutional codes over $\mathbb{Z}_{p^r}$, where $p$ is a prime. This addresses a conjecture from \cite{k}, which posits that every such code admits a minimal $p$-encoder,…

In this paper, we consider the Reed-Muller (RM) codes. For the first order RM code, we prove that it is unique in the sense that any linear code with the same length, dimension and minimum distance must be the first order RM code; For the…

For positive integers $s$ and $k_1, k_2, ..., k_s$, let $w(k_1,k_2,...,k_s)$ be the minimum integer $n$ such that any $s$-coloring $\{1,2,...,n\} \to \{1,2,...,s\}$ admits a $k_i$-term arithmetic progression of color $i$ for some $i$, $1…

For an integer $q\ge 2$, a perfect $q$-hash code $C$ is a block code over $[q]:=\{1,\ldots,q\}$ of length $n$ in which every subset $\{\mathbf{c}_1,\mathbf{c}_2,\dots,\mathbf{c}_q\}$ of $q$ elements is separated, i.e., there exists…

The purpose of this paper is to study the cyclic self orthogonal codes over $\mathbb{Z}_{p^m}$. After providing the generator polynomial of cyclic self orthogonal codes over $\mathbb{Z}_{p^m}$, we give the necessary and sufficient condition…

A fundamental theorem of linear algebra asserts that every basis for the vector space $\mathbb{R}^n$ has $n$ elements. In this expository note we present a theorem of W. G. Leavitt describing one way in which this invariant basis number…

We investigate perfect codes in $\mathbb{Z}^n$ under the $\ell_p$ metric. Upper bounds for the packing radius $r$ of a linear perfect code, in terms of the metric parameter $p$ and the dimension $n$ are derived. For $p = 2$ and $n = 2, 3$,…

Universally decodable matrices (UDMs) can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers $L$ and $N$ and a prime power $q$. The main result of this paper is…

In this paper we investigate codes over finite commutative rings R, whose generator matrices are built from \$\alpha\$-circulant matrices. For a non-trivial ideal I<R we give a method to lift such codes over R/I to codes over R, such that…

A $q$-ary $t$-covering array is an $m \times n$ matrix with entries from $\{0, 1, ..., q-1\}$ with the property that for any $t$ column positions, all $q^t$ possible vectors of length $t$ occur at least once. One wishes to minimize $m$ for…

We give one more proof of the first linear programming bound for binary codes, following the line of work initiated by Friedman and Tillich. The new argument is somewhat similar to previous proofs, but we believe it to be both simpler and…