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In this paper, we give upper bounds on the sizes of $(d, L)$ list-decodable codes in the Hamming metric space from covering codes with the covering radius smaller than or equal to $d$. When the list size $L$ is $1$, this gives many new Singleton type upper bounds on the sizes of codes with a given minimum Hamming distance. These upper bounds are stronger than the Griesmer bound when the lengths of codes are large. Some upper bounds on the lengths of general small Singleton defect codes or list-decodable codes attaining the generalized Singleton bound are given. As an application of our generalized Singleton type upper bounds on Hamming metric error-correcting codes, the generalized Singleton type upper bounds on insertion-deletion codes are given, which are much stronger than the direct Singleton bound for insertion-deletion codes when the lengths are large. We also give upper bounds on the lengths of small dimension optimal locally recoverable codes and small dimension optimal $(r, \delta)$ locally recoverable codes with any fixed given minimum distance., Comment: 29 pages. arXiv admin note: text overlap with arXiv:2109.02818

Private information retrieval (PIR) schemes (with or without colluding servers) have been proposed for realistic coded distributed data storage systems. Star product PIR schemes with colluding servers for general coded distributed storage system were constructed over general finite fields by R. Freij-Hollanti, O. W. Gnilke, C. Hollanti and A. Karpuk in 2017. These star product PIR schemes with colluding servers are suitable for the storage of files over small fields and can be constructed for coded distributed storage system with large number of servers. In this paper for an efficient storage code, the problem to find good retrieval codes is considered. In general if the storage code is a binary Reed-Muller code the retrieval code needs not to be a binary Reed-Muller code in general. It is proved that when the storage code contains some special codewords, nonzero retrieval rate star product PIR schemes with colluding servers can only protect against small number of colluding servers. We also give examples to show that when the storage code is a good cyclic code, the best choice of the retrieval code is not cyclic in general. Therefore in the design of star product PIR schemes with colluding servers, the scheme with the storage code and the retrieval code in the same family of algebraic codes is not always efficient., Comment: 25 pages,PIR schemes with the storage code and the retrieval code in the same family of algebraic codes seem not always efficient. arXiv admin note: text overlap with arXiv:2207.03163

An analysis of precession fishbone, diamagnetic fishbone and internal kink mode in Tokamak plasmas is presented via solving the fishbone dispersion relation. Applying the dispersion relation to a typical EAST discharge, excitation of precession fishbone due to Neutral Beam Injection is successfully explained. The real frequency and growth rate of diamagnetic fishbone and internal kink mode are calculated, and the relevance of the diamagnetic branch is also discussed for the possible equilibrium profile.

A basic problem for the constant dimension subspace coding is to determine the maximal possible size A_q (n, d, k) of a set of k-dimensional subspaces in Fnq such that the subspace distance satisfies d(U, V )> or =d for any two different subspaces U andV in this set. We present two new constructions of constant dimension subspace codes using subsets of maximal rank-distance (MRD) codes in several blocks. This method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting MRD codes. In these two constructions, subsets of MRD codes with bounded ranks play an essential role. The Delsarte theorem of the rank distribution of MRD codes is an important ingredient to count codewords in our constructed constant dimension subspace codes. We give many new lower bounds for A_q (n, d, k). More than 110 new constant dimension subspace codes better than previously best known codes are constructed., Comment: 22 pages, 5 tables

Constructions of quantum MDS codes have been studied by many authors. We refer to the table in page 1482 of [3] for known constructions. However there are only few $q$-ary quantum MDS $[[n,n-2d+2,d]]_q$ codes with minimum distances $d>\frac{q}{2}$ for sparse lengths $n>q+1$. In the case $n=\frac{q^2-1}{m}$ where $m|q+1$ or $m|q-1$ there are complete results. In the case $n=\frac{q^2-1}{m}$ where $m|q^2-1$ is not a factor of $q-1$ or $q+1$, there is no $q$-ary quantum MDS code with $d> \frac{q}{2}$ has been constructed. In this paper we propose a direct approch to construct Hermitian self-orthogonal codes over ${\bf F}_{q^2}$. Thus we give some new $q$-ary quantum codes in this case. Moreover we present many new $q$-ary quantum MDS codes with lengths of the form $\frac{w(q^2-1)}{u}$ and minimum distances $d > \frac{q}{2}$., Comment: 19 pages, submitted

The expicit restricted isometry property (RIP) measurement matrices are needed in practical application of compressed sensing in signal processing. RIP matrices from Reed-Solomon codes, BCH codes, orthogonal codes, expander graphs have been proposed and analysised. On the other hand binary constant weight codes have been studied for many years and many optimal or near-optimal small weight and ditance constant weight codes have been determined. In this paper we propose a new deterministic construction of RIP measurement matrices in compressed sensing from binary and ternary contant weight codes. The sparse orders and the number of budged rows in the new constant-weight-code-based RIP matrices can be arbitrary. These contant-weight-code based RIP matrices have better parameters compared with the DeVore RIP matrices when the sizes are small., Comment: 11 pages, submitted

Binary constant weight codes have important applications and have been studied for many years. Optimal or near-optimal binary constant weight codes of small lengths have been determined. In this paper we propose a new construction of explicit binary constant weight codes from $q$-ary Reed-Solomon codes. Some of our binary constant weight codes are optimal or new. In particular new binary constant weight codes $A(64, 10, 8) \geq 4108$ and $A(64, 12, 8) \geq 522$ are constructed. We also give explicitly constructed binary constant weight codes which improve Gilbert and Graham-Sloane lower bounds in some range of parameters. An extension to algebraic geometric codes is also presented., Comment: 15 pages, submitted