There are exactly two non-commutative rings of size 4, namely,$E = \langle a, b \vert 2a = 2b = 0, a^{2} = a, b^{2} = b, ab= a, ba = b\rangle $and its opposite ring$F$. These rings are non-unital. A subset$D$of$E^{m}$is defined with the help of simplicial complexes, and utilized to construct the linear left-$E$-code$C^{L}_{D}=\{(v\cdot d)_{d\in D}: v\in E^{m}\}$and the right-$E$-code$C^{R}_{D}=\{(d\cdot v)_{d\in D}: v\in E^{m}\}$. We study a certain binary subfield-like code corresponding to$C_{D}^{L}$. By using a Gray map, we also obtain the binary Gray images of$C_{D}^{L}$and$C_{D}^{R}$. The weight distributions of all these codes are computed. We achieve a couple of infinite families of optimal codes with respect to the Griesmer bound. Ashikhmin-Barg’s condition for minimality of a linear code is satisfied by most of the binary codes we constructed here. All the binary codes in this article are self-orthogonal and few-weight codes under certain mild conditions. This is the first attempt to study the structure of linear codes over a non-unital non-commutative ring using simplicial complexes.

The advent of 5G networking technologies has increased the expectations from mobile devices, in that, more sophisticated, computationally intense applications are expected to be delivered on the mobile device which are themselves getting smaller and sleeker. This predicates a need for offloading computationally intense parts of the applications to a resource strong cloud. Parallely, in the wireless networking world, the trend has shifted to multi-radio (as opposed to multi-channel) enabled communications. In this paper, we provide a comprehensive computation offloading solution that uses the multiple radio links available for associated data transfer, optimally. Our contributions include: a comprehensive model for the energy consumption from the perspective of the mobile device; the formulation of the joint optimization problem to minimize the energy consumed as well as allocating the associated data transfer optimally through the available radio links and an iterative algorithm that converges to a locally optimal solution. Simulations on an HTC phone, running a 14-component application and using the Amazon EC2 as the cloud, show that the solution obtained through the iterative algorithm consumes only 3% more energy than the optimal solution (obtained via exhaustive search).