Properties and classifications of certain LCD codes.

A linear code $C$ is called a linear complementary dual code (LCD code) if $C \cap C^\perp = {0}$ holds. LCD codes have many applications in cryptography, communication systems, data storage, and quantum coding theory. In this dissertation we show that a necessary and sufficient condition for a cyclic code $C$ over $\Z_4$ of odd length to be an LCD code is that $C=\big( f(x) \big)$ where $f$ is a self-reciprocal polynomial in $\Z_{4}[X]$ which is also in our paper \cite{GK1}. We then extend this result and provide a necessary and sufficient condition for a cyclic code $C$ of length $N$ over a finite chain ring R=\big(R,\m=(\gamma),\kappa=R/\m \big) with $\nu(\gamma)=2$ to be an LCD code. In \cite{DKOSS} a linear programming bound for LCD codes and the definition for $\text{LD}_{2}(n, k)$ for binary LCD $[n, k]$-codes are provided. Thus, in a different direction, we find the formula for $\text{LD}_{2}(n, 2)$ which appears in \cite{GK2}. In 2020, Pang et al. defined binary $\text{LCD}\; [n,k]$ codes with biggest minimal distance, which meets the Griesmer bound \cite{Pang}. We give a correction to and provide a different proof for \cite[Theorem 4.2]{Pang}, provide a different proof for \cite[Theorem 4.3]{Pang}, examine properties of LCD ternary codes, and extend some results found in \cite{Harada} for any $q$ which is a power of an odd prime

On the Structure of the Linear Codes with a Given Automorphism

The purpose of this paper is to present the structure of the linear codes over a finite field with q elements that have a permutation automorphism of order m. These codes can be considered as generalized quasi-cyclic codes. Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail, presenting necessary and sufficient conditions for which linear codes with such an automorphism are self-orthogonal, self-dual, or linear complementary dual

Characterization and mass formulas of symplectic self-orthogonal and LCD codes and their application

The object of this paper is to study two very important classes of codes in coding theory, namely self-orthogonal (SO) and linear complementary dual (LCD) codes under the symplectic inner product, involving characterization, constructions, and their application. Using such a characterization, we determine the mass formulas of symplectic SO and LCD codes by considering the action of the symplectic group, and further obtain some asymptotic results. Finally, under the Hamming distance, we obtain some symplectic SO (resp. LCD) codes with improved parameters directly compared with Euclidean SO (resp. LCD) codes. Under the symplectic distance, we obtain some additive SO (resp. additive complementary dual) codes with improved parameters directly compared with Hermitian SO (resp. LCD) codes. Further, we also construct many good additive codes outperform the best-known linear codes in Grassl's code table. As an application, we construct a number of record-breaking (entanglement-assisted) quantum error-correcting codes, which improve Grassl's code table

Embedding and decoding hidden data channels on computer displays

In a given setup, a LCD screen is mounted horizontally so that objects, e.g. game tokens, can be placed on the surface of the display. A PC displays information on the screen, e.g. a floor plan of a board game. The object placed on the screen is equipped with one or more optical sensors that ¿look¿ at the display area under the object. The information captured by the sensor is transmitted back to the PC that displays the information (after optional pre-processing the object). In order to determine de position of the objects in the screen, the screen is partitioned into a large number of separate areas. A separate hidden data channel has to be implemented in each of these areas. The data transmitted through these data channels can be used to transmit position identifying the given area. The task of this work is to research and design a system for implementing such a multitude of hidden data channels on a video display with the following constraints: - The displayed hidden data channel shall be invisible or at least unobtrusive to the human eye in the chosen setup. - The capacity of the channel should be as large as possible. - The system should be able to function if the sensor is placed between two, or three of the hidden data channel areas. - The system needs to be robust against typical error signals like ¿strobing back light of the display¿. - The system should be able to adapt to special conditions found on different types of display families.Querol Giner, AJ. (2009). Embedding and decoding hidden data channels on computer displays. http://hdl.handle.net/10251/21015.Archivo delegad