Phase Precoding for the Compute-and-Forward Protocol

The compute-and-forward (CoF) is a relaying protocol, which uses algebraic structured codes to harness the interference and remove the noise in wireless networks. We propose the use of phase precoders at the transmitters of a network, where relays apply CoF strategy. We define the {em phase precoded computation rate} and show that it is greater than the original computation rate of CoF protocol. We further give a new low-complexity method for finding network equations. We finally show that the proposed precoding scheme increases the degrees-of-freedom (DoF) of CoF protocol. This overcomes the limitations on the DoF of the CoF protocol, recently presented by Niesen and Whiting. Using tools from Diophantine approximation and algebraic geometry, we prove the existence of a phase precoder that approaches the maximum DoF when the number of transmitters tends to infinity.

Phase Precoded Compute-and-Forward with Partial Feedback

In this work, we propose phase precoding for the compute-and-forward (CoF) protocol. We derive the phase precoded computation rate and show that it is greater than the original computation rate of CoF protocol without precoder. To maximize the phase precoded computation rate, we need to jointly find the optimum phase precoding matrix and the corresponding network equation coefficients. This is a mixed integer programming problem where the optimum precoders should be obtained at the transmitters and the network equation coefficients have to be computed at the relays. To solve this problem, we introduce phase precoded CoF with partial feedback. It is a quantized precoding system where the relay jointly computes both a quasi-optimal precoder from a finite codebook and the corresponding network equations. The index of the obtained phase precoder within the codebook will then be fedback to the transmitters. A deep hole phase precoder is presented as an example of such a scheme. We further simulate our scheme with a lattice code carved out of the Gosset lattice and show that significant coding gains can be obtained in terms of equation error performance.

On the performance of 1-level LDPC lattices

The low-density parity-check (LDPC) lattices perform very well in high dimensions under generalized min-sum iterative decoding algorithm. In this work we focus on 1-level LDPC lattices. We show that these lattices are the same as lattices constructed based on Construction A and low-density lattice-code (LDLC) lattices. In spite of having slightly lower coding gain, 1-level regular LDPC lattices have remarkable performances. The lower complexity nature of the decoding algorithm for these type of lattices allows us to run it for higher dimensions easily. Our simulation results show that a 1-level LDPC lattice of size 10000 can work as close as 1.1 dB at normalized error probability (NEP) of $10^{-5}$.This can also be reported as 0.6 dB at symbol error rate (SER) of $10^{-5}$ with sum-product algorithm.

Turbo Lattices: Construction and Error Decoding Performance

In this paper a new class of lattices called turbo lattices is introduced and established. We use the lattice Construction D to produce turbo lattices. This method needs a set of nested linear codes as its underlying structure. We benefit from turbo codes as our basis codes. Therefore, a set of nested turbo codes based on nested interleavers (block interleavers) and nested convolutional codes is built. To this end, we employ both tail-biting and zero-tail convolutional codes. Using these codes, along with construction D, turbo lattices are created. Several properties of Construction D lattices and fundamental characteristics of turbo lattices including the minimum distance, coding gain and kissing number are investigated. Furthermore, a multi-stage turbo lattice decoding algorithm based on iterative turbo decoding algorithm is given. We show, by simulation, that turbo lattices attain good error performance within $sim1.25 dB$ from capacity at block length of $n=1035$. Also an excellent performance of only $sim.5 dB$ away from capacity at SER of $10^{-5}$ is achieved for size $n=10131$.

Cycle structure of permutation functions over finite fields and their applications

In this work we establish some new interleavers based on permutation functions. The inverses of these interleavers are known over a finite field $mathbb{F}_q$. For the first time M{o}bius and Redei functions are used to give new deterministic interleavers. Furthermore we employ Skolem sequences in order to find new interleavers with known cycle structure. In the case of Redei functions an exact formula for the inverse function is derived. The cycle structure of Redei functions is also investigated. The self-inverse and non-self-inver

Capacity Achieving Low Density Parity Check Lattices

The concept and existence of sphere-bound-achieving and capacity-achieving lattices has been explained on AWGN channels by Forney. LDPC lattices, introduced by Sadeghi, perform very well under iterative decoding algorithm. In this work, we focus on an ensemble of regular LDPC lattices. We produce and investigate an ensemble of LDPC lattices with known properties. It is shown that these lattices are sphere-bound-achieving and capacity-achieving. As byproducts we find the minimum distance, coding gain, kissing number and an upper bound for probability of error for this special ensemble of regular LDPC lattices.