A code C is Z 2 Z 4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). In this paper Z 2 Z 4 -additive codes are studied. Their corresponding binary images, via the Gray map, are Z 2 Z 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. In order to do this, the appropriate concept of duality for Z 2 Z 4 -additive codes is defined and the parameters of their dual codes are computed.

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In the traditional two-rack distributed storage system (DSS) model, due to the assumption that the storage capacity of each node is the same, the minimum bandwidth regenerating (MBR) point becomes infeasible. In this paper, we design a new non-homogeneous two-rack model by proposing a generalization of the threshold function used to compute the tradeoff curve. We prove that by having the nodes in the rack with higher regenerating bandwidth stores more information, all the points on the tradeoff curve, including the MBR point, become feasible. Finally, we show how the non-homogeneous two-rack model outperforms the traditional model in the tradeoff curve between the storage per node and the repair bandwidth.

A code C is Z 2 Z 4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of Z 2 Z 4 -additive codes under an extended Gray map are called Z 2 Z 4 -linear codes. In this paper, the invariants for Z 2 Z 4 -linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of Z 2 Z 4 -linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a Z 2 Z 4 -linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a Z 2 Z 4 -linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a Z 2 Z 4 -linear code for each possible pair (r, k) is given.

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