Related papers: On Achievability of an $(r,l)$ Fractional Linear N…
It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vector linear solution when message dimension is $2$ [3]. It is not known whether this result can be generalized for an…
For each integer $m \geq 2$, a network is constructed which is solvable over an alphabet of size $m$ but is not solvable over any smaller alphabets. If $m$ is composite, then the network has no vector linear solution over any $R$-module…
It is known a vector linear solution may exist if and only if the characteristic of the finite field belongs to a certain set of primes. But, can increasing the message dimension make a network vector linearly solvable over a larger set of…
It is known that there exists a multiple-unicast network which has a rate $1$ linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes. In this paper, we show…
We prove the following results regarding the linear solvability of networks over various alphabets. For any network, the following are equivalent: (i) vector linear solvability over some finite field, (ii) scalar linear solvability over…
We consider the following feasibility problem: given an integer $n \geq 1$ and an integer $m$ such that $0 \leq m \leq \binom{n}{2}$, does there exist a line graph $L = L(G)$ with exactly $n$ vertices and $m$ edges ? We say that a pair…
A linear Fractional Network Coding (FNC) solution over $\mathbb{F}_q$ is a linear network coding solution over $\mathbb{F}_q$ in which the message dimensions need not necessarily be the same and need not be the same as the edge vector…
Vector linear network coding (LNC) is a generalization of the conventional scalar LNC, such that the data unit transmitted on every edge is an $L$-dimensional vector of data symbols over a base field GF($q$). Vector LNC enriches the choices…
A directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources. We call such a network a sum-network. It is shown that there exists a solvably (and linear solvably)…
Sum-networks are networks where all the terminals demand the sum of the symbols generated at the sources. It has been shown that for any finite set/co-finite set of prime numbers, there exists a sum-network which has a vector linear…
For which positive integers $n,k,r$ does there exist a linear $[n,k]$ code $C$ over $\mathbb{F}_q$ with all codeword weights divisible by $q^r$ and such that the columns of a generating matrix of $C$ are projectively distinct? The…
Recent work of C. Fefferman and the first author has demonstrated that the linear system of equations \begin{equation*} \sum_{j=1}^M A_{ij}(x)F_j(x)=f_i(x)\hspace{.2in} (i=1,...,N), \end{equation*} has a $C^m$ solution $F=(F_1,...,F_M)$ if…
In an acyclic multicast network, it is well known that a linear network coding solution over GF($q$) exists when $q$ is sufficiently large. In particular, for each prime power $q$ no smaller than the number of receivers, a linear solution…
In [8], K. Kaur, S. Kamada et al. posed a problem of finding a virtual knot, if exists, with an unknotting index (n,m), where (n,m) is a pair of non-negative integers. In this paper, we address this question by providing infinite families…
Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a $k$-dimensional linear code over $\mathbb{F}_q$…
Fixed-size commutative rings are quasi-ordered such that all scalar linearly solvable networks over any given ring are also scalar linearly solvable over any higher-ordered ring. As consequences, if a network has a scalar linear solution…
Real world networks have, for a long time, been modelled by scale-free networks, which have many sparsely connected nodes and a few highly connected ones (the hubs). However, both in society and in biology, a new structure must be…
We demonstrate a kind of linear superposition for a large number of nonlinear equations, both continuum and discrete. In particular, we show that whenever a nonlinear equation admits solutions in terms of Jacobi elliptic functions…
It is known that for any finite/co-finite set of primes there exists a network which has a rate $1$ solution if and only if the characteristic of the finite field belongs to the given set. We generalize this result to show that for any…
We consider a two-unicast-$Z$ network over a directed acyclic graph of unit capacitated edges; the two-unicast-$Z$ network is a special case of two-unicast networks where one of the destinations has apriori side information of the unwanted…