Quantum MDS codes with new length and large minimum distance

According to the well-known CSS construction, constructing quantum MDS codes are extensively investigated via Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. In this paper, given two Hermitian self-orthogonal GRS codes $GR{S}_{k_1}(A,{\mathit{v}}_A)$ and $GR{S}_{k_2}(B,{\mathit{v}}_B)$, we propose a sufficient condition to ensure that $GR{S}_k(A\cup B,{\mathit{v}}_{A\cup B})$ is still a Hermitian self-orthogonal code. Consequently, we first present a new general construction of infinitely families of quantum MDS codes from known ones. Moreover, applying the trace function and norm function over finite fields, we give another two new constructions of quantum MDS codes with flexible parameters. It turns out that the forms of the lengths of our quantum MDS codes are quite different from previous known results in the literature. Meanwhile, the minimum distances of all the q-ary quantum MDS codes are bigger than $q/2+1$.     

Locally recoverable J-affine variety codes

A locally recoverable (LRC) code is a code over a finite field ${\mathbb{F}}_q$ such that any erased coordinate of a codeword can be recovered from a small number of other coordinates in that codeword. We construct LRC codes correcting more than one erasure, which are subfield-subcodes of some J-affine variety codes. For these LRC codes, we compute localities $(r,\delta )$ that determine the minimum size of a set $\bar{R}$ of positions so that any $\delta -1$ erasures in $\bar{R}$ can be recovered from the remaining r coordinates in this set. We also show that some of these LRC codes with lengths $n\gg q$ are $(\delta -1)$-optimal.     

Optimal cyclic (r,δ) locally repairable codes with unbounded length

Locally repairable codes with locality r (r-LRCs for short) were introduced by Gopalan et al. [1] to recover a failed node of the code from at most other r available nodes. And then $(r,\delta )$-locally repairable codes ($(r,\delta )$-LRCs for short) were produced by Prakash et al. [2] for tolerating multiple failed nodes. An r-LRC can be viewed as an $(r,2)$-LRC. An $(r,\delta )$-LRC is called optimal if it achieves the Singleton-type bound. It has been a great challenge to construct q-ary optimal $(r,\delta )$-LRCs with length much larger than q. Surprisingly, Luo et al. [3] presented a construction of q-ary optimal r-LRCs of minimum distances 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of q) via cyclic codes.In this paper, inspired by the work of [3], we firstly construct two classes of optimal cyclic $(r,\delta )$-LRCs with unbounded lengths and minimum distances $\delta +1$ or $\delta +2$, which generalize the results about the $\delta =2$ case given in [3]. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic $(r,\delta )$-LRCs with unbounded length and larger minimum distance 2δ. Furthermore, when $\delta =3$, we give another class of optimal cyclic $(r,3)$-LRCs with unbounded length and minimum distance 6.     

Galois LCD codes over mixed alphabets

In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of ${\mathbb{F}}_p{\mathbb{F}}_p[\theta ]$-linear codes, where $p\in \{2;3\}$ and $\theta \ne {\theta }^2=0$ that provides LCD codes over ${\mathbb{F}}_p$.     

Galois hulls of special Goppa codes and related codes with application to EAQECCs

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over ${\mathbb{F}}_{q^m}$ are GRS codes (resp., extended GRS codes) when $m=1$. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs.     

Construction of systematic (k + 1,k) memory MDS sliding window codes over embedded fields

In this paper, based on the structure of embedded fields, we investigate explicit construction of systematic $(k+1,k)$ mMDS sliding window codes with memory $m=2,3$. First, over GF($2^{lh}$) with $l\ge 1$ and $h\ge 2$, we propose an algorithm to construct $(2^{lh-1}+1,2^{lh-1})$ mMDS codes with memory 2, which are optimal in the sense that $2^{lh-1}$ is the maximum possible value of k for a $(k+1,k)$ sliding window code with memory 2 over GF($2^{lh}$) to be mMDS. When $l\ge 2$, every constructed code has the extra property that it contains a $(2^{h-1}+1,2^{h-1})$ mMDS sliding window code with memory 2 as a subcode over the subfield GF($2^h$). Next, over GF($2^{2lh}$) with $l\ge 1$ and $h\ge 2$, we introduce a method to construct $(k+1,k)$ mMDS codes memory 3, and a few new codes have been obtained consequently. When $l\ge 2$, every code constructed by the new approach also has the property that it contains an mMDS subcode over the subfield GF($2^{2h}$). The embedding subfield-subcode property enhances the flexibility and efficiency of the designed codes.     

Additive Hermitian idempotent preservers between operator algebras

Let L be an additive map between (real or complex) matrix algebras sending $n\times n$ Hermitian idempotent matrices to $m\times m$ Hermitian idempotent matrices. We show that there are nonnegative integers $p,q$ with $n(p+q)=r\le m$ and an $m\times m$ unitary matrix U such that$L(A)=U[(I_p?A)\oplus (I_q?A^t)\oplus 0_{m?r}]U^{?},\text{for any}n\times n\text{Hermitian}A\text{with rational trace}.$ We also extend this result to the (complex) von Neumann algebra setting, and provide a supplement to the Dye-Bunce-Wright Theorem asserting that every additive map of Hermitian idempotents extends to a Jordan ?-homomorphism.