Structural properties of binary propelinear codes
Joaquim Borges Ivan Yu. Mogilnykh Josep Rifà Faina I. Solov'eva
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.
keywords: transitive codes Propelinear codes binary perfect codes.
Families of nested completely regular codes and distance-regular graphs
Joaquim Borges Josep Rifà Victor A. Zinoviev
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.
keywords: Completely regular codes completely transitive codes distance-regular graphs distance-transitive graphs.
$\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in Steganography
Helena Rifà-Pous Josep Rifà Lorena Ronquillo
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.
keywords: $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes. Steganography
On $q$-ary linear completely regular codes with $\rho=2$ and antipodal dual
Joaquim Borges Josep Rifà Victor A. Zinoviev
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.
keywords: covering radius. Linear completely regular codes completely transitive codes
Completely regular codes by concatenating Hamming codes
Joaquim Borges Josep Rifà Victor Zinoviev

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.

keywords: Completely regular codes completely transitive codes concatenation extended codes intersection array

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