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In this paper, we study linear complementary pairs (LCP) of codes over finite non-commutative local rings. We further provide a necessary and sufficient condition for a pair of codes $(C,D)$ to be LCP of codes over finite non-commutative Frobenius rings. The minimum distances $d(C)$ and $d(D^\perp)$ are defined as the security parameter for an LCP of codes $(C, D).$ It was recently demonstrated that if $C$ and $D$ are both $2$-sided LCP of group codes over a finite commutative Frobenius rings, $D^\perp$ and $C$ are permutation equivalent in \cite{LL23}. As a result, the security parameter for a $2$-sided group LCP $(C, D)$ of codes is simply $d(C)$. Towards this, we deliver an elementary proof of the fact that for a linear complementary pair of codes $(C,D)$, where $C$ and $D$ are linear codes over finite non-commutative Frobenius rings, under certain conditions, the dual code $D^\perp$ is equivalent to $C.$

An additive code is an $\mathbb{F}_q$-linear subspace of $\mathbb{F}_{q^m}^n$ over $\mathbb{F}_{q^m}$, which is not a linear subspace over $\mathbb{F}_{q^m}$. Linear complementary pairs (LCP) of codes have important roles in cryptography, such as increasing the speed and capacity of digital communication and strengthening security by improving the encryption necessities to resist cryptanalytic attacks. This paper studies an algebraic structure of additive complementary pairs (ACP) of codes over $\mathbb{F}_{q^m}$. Further, we characterize an ACP of codes in analogous generator matrices and parity check matrices. Additionally, we identify a necessary condition for an ACP of codes. Besides, we present some constructions of an ACP of codes over $\mathbb{F}_{q^m}$ from LCP codes over $\mathbb{F}_{q^m}$ and also from an LCP of codes over $\mathbb{F}_{q}$. Finally, we study the constacyclic ACP of codes over $\mathbb{F}_{q^m}$ and the counting of the constacyclic ACP of codes.

Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of codes have been proposed for new applications as countermeasures against side-channel attacks (SCA) and fault injection attacks (FIA) in the context of direct sum masking (DSM). The countermeasure against FIA may lead to a vulnerability for SCA when the whole algorithm needs to be masked (in environments like smart cards). This led to a variant of the LCD and LCP problems, where several results have been obtained intensively for LCD codes, but only partial results have been derived for LCP codes. Given the gap between the thin results and their particular importance, this paper aims to reduce this by further studying the LCP of codes in special code families and, precisely, the characterisation and construction mechanism of LCP codes of algebraic geometry codes over finite fields. Notably, we propose constructing explicit LCP of codes from elliptic curves. Besides, we also study the security parameters of the derived LCP of codes $(\mathcal{C}, \mathcal{D})$ (notably for cyclic codes), which are given by the minimum distances $d(\mathcal{C})$ and $d(\mathcal{D}^\perp)$. Further, we show that for LCP algebraic geometry codes $(\mathcal{C},\mathcal{D})$, the dual code $\mathcal{C}^\perp$ is equivalent to $\mathcal{D}$ under some specific conditions we exhibit. Finally, we investigate whether MDS LCP of algebraic geometry codes exist (MDS codes are among the most important in coding theory due to their theoretical significance and practical interests). Construction schemes for obtaining LCD codes from any algebraic curve were given in 2018 by Mesnager, Tang and Qi in [``Complementary dual algebraic geometry codes", IEEE Trans. Inform Theory, vol. 64(4), 2390--3297, 2018]. To our knowledge, it is the first time LCP of algebraic geometry codes has been studied.

We provide a simple proof for a complementary pair of group codes over a finite non-commutative Frobenius ring of the fact that one of them is equivalent to the other one. We also explore this fact for checkeable codes over the same type of alphabet.

Generalizing the linear complementary duals, the linear complementary pairs and the hull of codes, we introduce the concept of $\ell$-dimension linear intersection pairs ($\ell$-DLIPs) of codes over a finite commutative ring $(R)$, for some positive integer $\ell$. In this paper, we study $\ell$-DLIP of codes over $R$ in a very general setting by a uniform method. Besides, we provide a necessary and sufficient condition for the existence of a non-free (or free) $\ell$-DLIP of codes over a finite commutative Frobenius ring. In addition, we obtain a generator set of the intersection of two constacyclic codes over a finite chain ring, which helps us to get an important characterization of $\ell$-DLIP of constacyclic codes. Finally, the $\ell$-DLIP of constacyclic codes over a finite chain ring are used to construct new entanglement-assisted quantum error correcting (EAQEC) codes.

We establish a complete classification of binary group codes with complementary duals for a finite group and explicitly determine the number of linear complementary dual (LCD) cyclic group codes by using cyclotomic cosets. The dimension and the minimum distance for LCD group codes are explored. Finally, we find a connection between LCD MDS group codes and maximal ideals., Comment: 11 pages

A code is called $(n, k, r, t)$ information symbol locally repairable code \big($(n, k, r, t)_i$ LRC\big) if each information coordinate can be achieved by at least $t$ disjoint repair sets, containing at most $r$ other coordinates. This paper considers a class of $(n, k, r, t)_i$ LRCs, where each repair set contains exactly one parity coordinate. We explore the systematic code in terms of the standard parity check matrix. First, some structural features of the parity check matrix are proposed by showing some connections with the membership matrix and the minimum distance optimality of the code. Next to that, parity check matrix based proofs of various bounds associated with the code are placed. In addition to this, we provide several constructions of optimal $(n, k, r, t)_i$ LRCs, with the help of two Cayley tables of a finite field. Finally, we generalize a result of $q$-ary $(n, k, r)$ LRCs to $q$-ary $(n, k, r, t)$ LRCs., Comment: 12 pages

Hiding a secret is needed in many situations. Secret sharing plays an important role in protecting information from getting lost, stolen, or destroyed and has been applicable in recent years. A secret sharing scheme is a cryptographic protocol in which a dealer divides the secret into several pieces of share and one share is given to each participant. To recover the secret, the dealer requires a subset of participants called access structure. In this paper, we present a multi-secret sharing scheme over a local ring based on linear complementary dual codes using Blakley's method. We take a large secret space over a local ring that is greater than other code-based schemes and obtain a perfect and almost ideal scheme., Comment: 12 pages

Graphite, with many industrial applications, is one of the widely sought-after allotropes of carbon. The sp2 hybridized and thermodynamically stable form of carbon forms a layered structure with strong in-plane carbon bonds and weak inter-layer van der Waals bonding. Graphite is also a high-temperature ceramic, and shaping them into complex geometries is challenging, given its limited sintering behavior even at high temperatures. Although the geometric design of the graphite structure in many of the applications could dictate its precision performance, conventional synthesis methods for formulating complex geometric graphite shapes are limited due to the intrinsic brittleness and difficulties of high-temperature processing. Here, we report the development of colloidal graphite ink from commercial graphite powders with reproducible rheological behavior that allows the fabrication of any complex architectures with tunable geometry and directionality via 3D printing at room temperature. The method is enabled via using small amounts of clay, another layered material, as an additive, allowing the proper design of the graphene ink and subsequent binding of graphite platelets during printing. Sheared layers of clay are easily able to flow, adapt, and interface with graphite layers forming strong binding between the layers and between particles that make the larger structures. The direct ink printing of complex 3D architectures of graphite without further heat treatments could lead to easy shape engineering and related applications of graphite at various length scales, including complex graphite molds or crucibles. The 3D printed complex graphitic structures exhibit excellent thermal, electrical, and mechanical properties, and the clay additive does not seem to alter these properties due to the excellent inter-layer dispersion and mixing within the graphite material., Comment: 19 pages, 4 figures

In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings. We later show that a free constacyclic code over finite chain ring is LCD if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible over finite chain rings. We illustrate the minimum Lee-distance of LCD codes over some finite commutative chain rings and demonstrate the results with examples. We also got some new optimal $\mathbb{Z}_4$ codes of different lengths {which are} cyclic LCD codes over $\mathbb{Z}_4$.

In this paper, we study the codes over the matrix ring over $\mathbb{Z}_4$, which is perhaps the first time the ring structure $M_2(\mathbb{Z}_4)$ is considered as a code alphabet. This ring is isomorphic to $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$, where $w$ is a root of the irreducible polynomial $x^2+x+1 \in \mathbb{Z}_2[x]$ and $U\equiv$ ${11}\choose{11}$. We first discuss the structure of the ring $M_2(\mathbb{Z}_4)$ and then focus on algebraic structure of cyclic codes and self-dual cyclic codes over $M_2(\mathbb{Z}_4)$. We obtain the generators of the cyclic codes and their dual codes. Few examples are given at the end of the paper., Comment: 10 pages

Low-density, highly porous graphene/graphene oxide (GO) based-foams have shown high performance in energy absorption applications, even under high compressive deformations. In general, foams are very effective as energy dissipative materials and have been widely used in many areas such as automotive, aerospace and biomedical industries. In the case of graphene-based foams, the good mechanical properties are mainly attributed to the intrinsic graphene and/or GO electronic and mechanical properties. Despite the attractive physical properties of graphene/GO based-foams, their structural and thermal stabilities are still a problem for some applications. For instance, they are easily degraded when placed in flowing solutions, either by the collapsing of their layers or just by structural disintegration into small pieces. Recently, a new and scalable synthetic approach to produce low-density 3D macroscopic GO structure interconnected with polydimethylsiloxane (PDMS) polymeric chains (pGO) was proposed. A controlled amount of PDMS is infused into the freeze-dried foam resulting into a very rigid structure with improved mechanical properties, such as tensile plasticity and toughness. The PDMS wets the graphene oxide sheets and acts like a glue bonding PDMS and GO sheets. In order to obtain further insights on mechanisms behind the enhanced mechanical pGO response we carried out fully atomistic molecular dynamics (MD) simulations. Based on MD results, we build up a structural model that can explain the experimentally observed mechanical behavior.

A fibrous herringbone-modified helicoidal architecture is identified within the exocuticle of an impact-resistant crustacean appendage. This previously unreported composite microstructure, which features highly textured apatite mineral templated by an alpha-chitin matrix, provides enhanced stress redistribution and energy absorption over the traditional helicoidal design under compressive loading. Nanoscale toughening mechanisms are also identified using high load nanoindentation and in-situ TEM picoindentation., Comment: 6 figures. Published in Advanced Materials 2016