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### Open Access Journals

AMC

The paper deals with some structural properties of propelinear binary
codes, in particular propelinear perfect binary codes. We consider the
connection of transitive codes with propelinear codes and show that
there exists a binary code, the Best code of length 10, size 40 and
minimum distance 4, which is transitive but not propelinear. We propose
several constructions of propelinear codes and introduce a new large
class of propelinear perfect binary codes, called normalized propelinear
perfect codes. Finally, based on the different values for the rank and
the dimension of the kernel, we give a lower bound on the number of
nonequivalent propelinear perfect binary codes.

AMC

In this paper infinite families of linear binary nested completely
regular codes are constructed. They have covering radius $\rho$
equal to $3$ or $4,$ and are $1/2^i$th parts, for
$i\in\{1,\ldots,u\}$ of binary (respectively, extended binary)
Hamming codes of length $n=2^m-1$ (respectively, $2^m$), where
$m=2u$. In the usual way, i.e., as coset graphs, infinite families
of embedded distance-regular coset graphs of diameter $D$ equal
to $3$ or $4$ are constructed. This gives antipodal covers of some
distance-regular and distance-transitive graphs. In some cases, the constructed codes are
also completely transitive and the corresponding coset
graphs are distance-transitive.

AMC

Steganography is an information hiding application which aims to hide secret data imperceptibly into a cover object. In this paper, we describe a novel coding method based on $\mathbb{Z}_2\mathbb{Z}_4$-additive codes in which data is embedded by distorting each cover symbol by one unit at most ($\pm 1$-steganography). This method is optimal and solves the problem encountered by the most efficient methods known today, concerning the treatment of boundary values. The performance of this new technique is compared with that of the mentioned methods and with the well-known rate-distortion upper bound to conclude that a higher payload can be obtained for a given distortion by using the proposed method.

AMC

We characterize all $q$-ary linear completely regular codes with covering radius $\rho=2$ when the dual codes are antipodal. These completely regular codes are extensions of linear completely regular codes with covering radius 1, which we also classify. For $\rho=2$, we give a list of all such codes known to us. This also gives the characterization of two weight linear antipodal codes. Finally, for a class of completely regular codes with covering radius $\rho=2$ and antipodal dual, some interesting properties on self-duality and lifted codes are pointed out.

AMC

We construct new families of completely regular codes by concatenation methods. By combining parity check matrices of cyclic Hamming codes, we obtain families of completely regular codes. In all cases, we compute the intersection array of these codes. As a result, we find some non-equivalent completely regular codes, over the same finite field, with the same parameters and intersection array. We also study when the extension of these codes gives completely regular codes. Some of these new codes are completely transitive.

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