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It is well known that quantum codes can be constructed by means of classical symplectic dual-containing codes. This paper considers a family of two-generators quasi-cyclic codes and derives sufficient conditions for these codes to be dual-containing. Then, a new method for constructing binary quantum codes is proposed. As an application, we construct 11 binary quantum codes that exceed the beak-known results. Further, another 40 new binary quantum codes are obtained by propagation rules, all of which improve the lower bound on the minimum distance.

We present the first algorithm for fully dynamic $k$-centers clustering in an arbitrary metric space that maintains an optimal $2+\epsilon$ approximation in $O(k \cdot \operatorname{polylog}(n,\Delta))$ amortized update time. Here, $n$ is an upper bound on the number of active points at any time, and $\Delta$ is the aspect ratio of the data. Previously, the best known amortized update time was $O(k^2\cdot \operatorname{polylog}(n,\Delta))$, and is due to Chan, Gourqin, and Sozio. We demonstrate that the runtime of our algorithm is optimal up to $\operatorname{polylog}(n,\Delta)$ factors, even for insertion-only streams, which closes the complexity of fully dynamic $k$-centers clustering. In particular, we prove that any algorithm for $k$-clustering tasks in arbitrary metric spaces, including $k$-means, $k$-medians, and $k$-centers, must make at least $\Omega(n k)$ distance queries to achieve any non-trivial approximation factor. Despite the lower bound for arbitrary metrics, we demonstrate that an update time sublinear in $k$ is possible for metric spaces which admit locally sensitive hash functions (LSH). Namely, we demonstrate a black-box transformation which takes a locally sensitive hash family for a metric space and produces a faster fully dynamic $k$-centers algorithm for that space. In particular, for a large class of metrics including Euclidean space, $\ell_p$ spaces, the Hamming Metric, and the Jaccard Metric, for any $c > 1$, our results yield a $c(4+\epsilon)$ approximate $k$-centers solution in $O(n^{1/c} \cdot \operatorname{polylog}(n,\Delta))$ amortized update time, simultaneously for all $k \geq 1$. Previously, the only known comparable result was a $O(c \log n)$ approximation for Euclidean space due to Schmidt and Sohler, running in the same amortized update time.

We consider a problem introduced by Feige, Gamarnik, Neeman, R\'acz and Tetali [2020], that of finding a large clique in a random graph $G\sim G(n,\frac{1}{2})$, where the graph $G$ is accessible by queries to entries of its adjacency matrix. The query model allows some limited adaptivity, with a constant number of rounds of queries, and $n^\delta$ queries in each round. With high probability, the maximum clique in $G$ is of size roughly $2\log n$, and the goal is to find cliques of size $\alpha\log n$, for $\alpha$ as large as possible. We prove that no two-rounds algorithm is likely to find a clique larger than $\frac{4}{3}\delta\log n$, which is a tight upper bound when $1\leq\delta\leq \frac{6}{5}$. For other ranges of parameters, namely, two-rounds with $\frac{6}{5}<\delta<2$, and three-rounds with $1\leq\delta<2$, we improve over the previously known upper bounds on $\alpha$, but our upper bounds are not tight. If early rounds are restricted to have fewer queries than the last round, then for some such restrictions we do prove tight upper bounds.

For a positive integer $b\ge2$, the $b$-symbol code is a new coding framework proposed to combat $b$-errors in $b$-symbol read channels. Especially, a $2$-symbol code is called a symbol-pair code. Remarkably, a classical maximum distance separable (MDS) code is also an MDS $b$-symbol code. Recently, for any MDS code $\mathcal{C}$, Ma and Luo determined the symbol-pair weight distribution for $\mathcal{C}$. In this paper, by calculating the number of codewords in $\mathcal{C}$ with special shape, we obtain the $b$-weight distribution for $\mathcal{C}$, and then generalize Theorem $1$ in \cite{ML}.

List-decoding and list-recovery are important generalizations of unique decoding that received considerable attention over the years. However, the optimal trade-off among list-decoding (resp. list-recovery) radius, list size, and the code rate are not fully understood in both problems. This paper takes a step towards this direction when the list size is a given constant and the alphabet size is large (as a function of the code length). We prove a new Singleton-type upper bound for list-decodable codes, which improves upon the previously known bound by roughly a factor of $1/L$, where $L$ is the list size. We also prove a Singleton-type upper bound for list-recoverable codes, which is to the best of our knowledge, the first such bound for list-recovery. We apply these results to obtain new lower bounds that are optimal up to a multiplicative constant on the list size for list-decodable and list-recoverable codes with rates approaching capacity. Moreover, we show that list-decodable \emph{nonlinear} codes can strictly outperform list-decodable linear codes. More precisely, we show that there is a gap for a wide range of parameters, which grows fast with the alphabet size, between the size of the largest list-decodable nonlinear code and the size of the largest list-decodable linear codes. This is achieved by a novel connection between list-decoding and the notion of sparse hypergraphs in extremal combinatorics. We remark that such a gap is not known to exist in the problem of unique decoding. Lastly, we show that list-decodability or recoverability of codes implies in some sense good unique decodability.

Locally repairable codes enables fast repair of node failure in a distributed storage system. The code symbols in a codeword are stored in different storage nodes, such that a disk failure can be recovered by accessing a small fraction of the storage nodes. The number of storage nodes that are contacted during the repair of a failed node is a parameter called locality. We consider locally repairable codes that can be locally recovered in the presence of multiple node failures. The punctured code obtained by removing the code symbols in the complement of a repair group is called a local code. We aim at designing a code such that all local codes have a prescribed minimum distance, so that any node failure can be repaired locally, provided that the total number of node failures is less than the tolerance parameter. We consider linear locally repairable codes defined over a finite field of size four. This alphabet has characteristic 2, and hence is amenable to practical implementation. We classify all quaternary locally repairable codes that attain the Singleton-type upper bound for minimum distance. For each combination of achievable code parameters, an explicit code construction is given.