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 $iin{1,ldots,u}$ of binary (respectively, extended binary) Hamming co
des 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. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive.
In this paper new infinite families of linear binary completely transitive codes are presented. They have covering radius $rho = 3$ and 4, and are a half part of the binary Hamming and the binary extended Hamming code of length $n=2^m-1$ and $2^m$, r
espectively, where $m$ is even. From these new completely transitive codes, in the usual way, i.e., as coset graphs, new presentations of infinite families of distance transitive coset graphs of diameter three and four, respectively, are constructed.
Self-dual codes over $Z_2timesZ_4$ are subgroups of $Z_2^alpha timesZ_4^beta$ that are equal to their orthogonal under an inner-product that relates to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible
values $alpha,beta$ such that there exist a code $Csubseteq Z_2^alpha timesZ_4^beta$ are established. Moreover, the construction of a $add$-linear code for each type and possible pair $(alpha,beta)$ is given. Finally, the standard techniques of invariant theory are applied to describe the weight enumerators for each type.
Let C be an additive subgroup of $Z_{2k}^n$ for any $kgeq 1$. We define a Gray map $Phi:Z_{2k}^n longrightarrow Z_2^{kn}$ such that $Phi(codi)$ is a binary propelinear code and, hence, a Hamming-compatible group code. Moreover, $Phi$ is the unique Gr
ay map such that $Phi(C)$ is Hamming-compatible group code. Using this Gray map we discuss about the nonexistence of 1-perfect binary mixed group code.
Completely regular codes with covering radius $rho=1$ must have minimum distance $dleq 3$. For $d=3$, such codes are perfect and their parameters are well known. In this paper, the cases $d=1$ and $d=2$ are studied and completely characterized when t
he codes are linear. Moreover, it is proven that all these codes are completely transitive.
A code ${cal C}$ is $Z_2Z_4$-additive if the set of coordinates can be partitioned into two subsets $X$ and $Y$ such that the punctured code of ${cal C}$ by deleting the coordinates outside $X$ (respectively, $Y$) is a binary linear code (respectivel
y, a quaternary linear code). In this paper $Z_2Z_4$-additive codes are studied. Their corresponding binary images, via the Gray map, are $Z_2Z_4$-linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity check matrices are given. For this, the appropriate inner product is deduced and the concept of duality for $Z_2Z_4$-additive codes is defined. Moreover, the parameters of the dual codes are computed. Finally, some conditions for self-duality of $Z_2Z_4$-additive codes are given.
