A tight upper bound on the number of non-zero weights of a quasi-cyclic code

Let $\mathcal{C}$ be a quasi-cyclic code of index $l(l\geq2)$. Let $G$ be the subgroup of the automorphism group of $\mathcal{C}$ generated by $\rho^l$ and the scalar multiplications of $\mathcal{C}$, where $\rho$ denotes the standard cyclic shift. In this paper, we find an explicit formula of orbits of $G$ on $\mathcal{C}\setminus \{\mathbf{0}\}$. Consequently, an explicit upper bound on the number of non-zero weights of $\mathcal{C}$ is immediately derived and a necessary and sufficient condition for codes meeting the bound is exhibited. In particular, we list some examples to show the bounds are tight. Our main result improves and generalizes some of the results in \cite{M2}

Discovering Organizational Correlations from Twitter

Organizational relationships are usually very complex in real life. It is difficult or impossible to directly measure such correlations among different organizations, because important information is usually not publicly available (e.g., the correlations of terrorist organizations). Nowadays, an increasing amount of organizational information can be posted online by individuals and spread instantly through Twitter. Such information can be crucial for detecting organizational correlations. In this paper, we study the problem of discovering correlations among organizations from Twitter. Mining organizational correlations is a very challenging task due to the following reasons: a) Data in Twitter occurs as large volumes of mixed information. The most relevant information about organizations is often buried. Thus, the organizational correlations can be scattered in multiple places, represented by different forms; b) Making use of information from Twitter collectively and judiciously is difficult because of the multiple representations of organizational correlations that are extracted. In order to address these issues, we propose multi-CG (multiple Correlation Graphs based model), an unsupervised framework that can learn a consensus of correlations among organizations based on multiple representations extracted from Twitter, which is more accurate and robust than correlations based on a single representation. Empirical study shows that the consensus graph extracted from Twitter can capture the organizational correlations effectively.Comment: 11 pages, 4 figure

How many weights can a cyclic code have ?

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), {whose nonzero codewords have period equal to the length of the code}. Asymptotics are derived on the function $\Gamma(k,q),$ {that is defined as} the largest number of nonzero weights a cyclic code of dimension $k$ over \F_q can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or $q$-ary, as well as in the $q$-ary Hamming code are determined, two difficult results of independent interest.Comment: submitted on 21 June, 201

On the number of frequency hypercubes Fn(4;2,2)F^n(4;2,2)

A frequency $n$-cube $F^n(4;2,2)$ is an $n$-dimensional $4\times\cdots\times 4$ array filled by $0$s and $1$s such that each line contains exactly two $1$s. We classify the frequency $4$-cubes $F^4(4;2,2)$, find a testing set of size $25$ for $F^3(4;2,2)$, and derive an upper bound on the number of $F^n(4;2,2)$. Additionally, for any $n$ greater than $2$, we construct an $F^n(4;2,2)$ that cannot be refined to a latin hypercube, while each of its sub-$F^{n-1}(4;2,2)$ can. Keywords: frequency hypercube, frequency square, latin hypercube, testing set, MDS cod

Quasi-cyclic perfect codes in Doob graphs and special partitions of Galois rings

The Galois ring GR$(4^\Delta)$ is the residue ring $Z_4[x]/(h(x))$, where $h(x)$ is a basic primitive polynomial of degree $\Delta$ over $Z_4$. For any odd $\Delta$ larger than $1$, we construct a partition of GR$(4^\Delta) \backslash \{0\}$ into $6$-subsets of type $\{a,b,-a-b,-a,-b,a+b\}$ and $3$-subsets of type $\{c,-c,2c\}$ such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^\Delta)$ and, if $\Delta$ is not a multiple of $3$, under the action of the automorphism group of GR$(4^\Delta)$. As a corollary, this implies the existence of quasi-cyclic additive $1$-perfect codes of index $(2^\Delta-1)$ in $D((2^\Delta-1)(2^\Delta-2)/{6}, 2^\Delta-1 )$ where $D(m,n)$ is the Doob metric scheme on $Z^{2m+n}$.Comment: Accepted version; 7 IEEE TIT page

Sharp kinetic acceleration potentials during mediated redox catalysis of insulators

Redox mediators could catalyse otherwise slow and energy-inefficient cycling of Li-S and Li-O 2 batteries by shuttling electrons/holes between the electrode and the solid insulating storage materials. For mediators to work efficiently they need to oxidize the solid with fast kinetics yet the lowest possible overpotential. Here, we found that when the redox potentials of mediators are tuned via, e.g., Li + concentration in the electrolyte, they exhibit distinct threshold potentials, where the kinetics accelerate several-fold within a range as small as 10 mV. This phenomenon is independent of types of mediators and electrolyte. The acceleration originates from the overpotentials required to activate fast Li + /e – extraction and the following chemical step at specific abundant surface facets. Efficient redox catalysis at insulating solids requires therefore carefully considering the surface conditions of the storage materials and electrolyte-dependent redox potentials, which may be tuned by salt concentrations or solvents