This paper introduces a novel framework for constructing algebraic lattices based on Construction-D, leveraging nested linear codes and prime ideals from algebraic number fields. We focus on the application of these lattices in block-fading (BF) channels, which are characterized by piecewise-constant fading across blocks of transmitted symbols. This approach results in a semi-systematic generator matrix, providing a structured foundation for high -dimensional lattice design for BF channels. The proposed Construction-D lattices exhibit the full diversity property, making them highly effective for error performance improvement. To address this, we develop an efficient decoding algorithm designed specifically for full-diversity Construction-D lattices. Simulations indicate that the proposed lattices notably enhance error performance compared to full-diversity Construction-A lattices, especially in high-diversity scenarios. Interestingly, the anticipated performance gain from increasing the number of nested code levels was only observed in the two-level case. We believe that this is due to the amplified impact of error propagation during successive cancellation at higher levels. These findings highlight the promise of Construction-D lattices as an effective and practical coding strategy for enhancing communication reliability in BF channels.

This paper presents a novel framework for constructing full-diversity algebraic lattices based on Construction-D, utilizing nested linear codesC1⊂ · · · ⊂Ca⊆ FpNand prime ideals from algebraic number fields of degree n. Focused on block-fading (BF) channels, this approach yields a semi-systematic generator matrix suited for high-dimensional lattice design. The resulting Construction-D lattices achieve full diversity, significantly enhancing error performance. Additionally, we develop a decoding algorithm tailored for these full-diversity lattices, achieving linear complexity relative to the lattice dimension. Simulations indicate that the proposed lattices notably enhance error performance compared to full-diversity Construction-A lattices in diversity-2 cases. Interestingly, unlike AWGN channels, the expected performance enhancement of Construction-D over Construction-A, resulting from an increased number of nested code levels, was observed only in the two-level and diversity-2 cases. This phenomenon is likely due to the compounded error propagation during successive cancellation at higher levels, further amplified by higher diversity orders. The omission of parity-check enforcement at the final code level—aimed at simplifying our decoding—restricts the diversity order from the maximum ofn·dmin(Ca) ton, limiting performance gains. Unraveling the core reasons behind this decoding behavior remains an open challenge. Nevertheless, these findings highlight the promise of full-diversity Construction-D lattices as an effective coding strategy for BF channels.

Locally recoverable codes (LRCs) play a vital role in distributed storage systems where the failure or unavailability of storage devices is a common occurrence. The purpose of LRCs is to facilitate the repair processes required to recover lost or damaged data in such systems. A code C will be said (r, δ)-LRC if for each i, the ith component of codewords have locality (r, δ), that is, there exists a punctured subcode of C with support containing i, whose length is at most r + δ-1, and whose minimum distance is at least δ. An (r, δ)-LRC with locality (r, δ) allows for the local recovery of any δ-1 nodes by accessing information from r other nodes. In this paper, we present new constructions of (r, δ)-LRCs, with 2 ≤ δ ≤ p-1/t over Zps, where t divides p-1 and t ≠ p - 1. Initially, we provide generator matrices for (r, 2)-LRCs, among which one instance is considered as Singleton-Type Bound (STB)-optimal, a notion introduced in this paper. Also, we present a method for recovering an erased symbol in a codeword of our (r, 2)-LRC. For this aim, we use the polynomial interpolation over Zpsproposed by Gopalan. Next, we present the parity-check matrices for another family of (r, δ)-LRCs over Zps, and construct two instances of STB-optimal (r, δ)-LRCs. To the best of our knowledge, this paper presents the first study on ring-based LRCs. The proposed LRCs over Zpsexhibit certain design restrictions compared to LRCs over Fps. However, we provide two advantages for LRCs over Zps. First, we analyze Boolean circuits for arithmetic operations and demonstrate that the complexity of implementing multiplication in Zps, the operation with the highest cost in our algorithms, is considerably lower than in Fps. This highlights the superior performance of our LRCs in terms of implementation speed and cost-efficiency compared to their counterparts. Next, we offer an example illustrating that the Gray image of particular Zps+1-LRCs of lengthnresults in LRCs of lengthnpsover Fp, which may not necessarily be linear. This introduces a novel class of LRCs, prompting further exploration of the connections between existing nonlinear LRCs in finite fields and linear ring-based LRCs.

Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. Frequency-hopping schemes and orthogonal frequency division multiplexing (OFDM) can conveniently be modelled as BF channels. Applying lattices in this type of channel entails dividing a lattice point into multiple blocks such that fading is constant within a block but changes, independently, across blocks. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. Recently, the original binary Construction A for lattices, due to Forney, has been generalized to a lattice construction from totally real and complex multiplication (CM) fields. This generalized algebraic Construction A of lattices provides signal space diversity, intrinsically, which is the main requirement for the signal sets designed for fading channels. In this paper, we construct full-diversity algebraic lattices for BF channels using Construction A over totally real number fields. We propose two new decoding methods for these family of lattices which have complexity that grows linearly in the dimension of the lattice. The first decoder is proposed for full-diversity algebraic LDPC lattices which are generalized Construction A lattices with a binary LDPC code as underlying code. This decoding method contains iterative and non-iterative phases. In order to implement the iterative phase of our decoding algorithm, we propose the definition of a parity-check matrix and Tanner graph for full-diversity algebraic Construction A lattices. We also prove that using an underlying LDPC code that achieves the outage probability limit over one-block-fading channel, the constructed algebraic LDPC lattices together with the proposed decoding method admit diversity order $n$ over an $n$ -block-fading channel. Then, we modify the proposed algorithm by removing its iterative phase which enables full-diversity practical decoding of all generalized Construction A lattices without any assumption about their underlying code. In contrast with the known results on AWGN channels in which non-binary Construction A lattices always outperform the binary ones, we provide some instances showing that algebraic Construction A lattices obtained from binary codes outperform the ones based on non-binary codes in block fading channels. Since available lattice construction methods from totally real and complex multiplication (CM) fields do not provide diversity in the binary case, we generalize algebraic Construction A lattices over a wider family of number fields namely monogenic number fields.

We propose a new nonlinear Rao-Nam like symmetric key encryption scheme. In our design, we employ a specific type of coded modulation schemes namely quasi-cyclic low-density parity-check (QC-LDPC) lattice-codes which have low-complexity encoding and decoding algorithms. Due to the application of coded modulation schemes in our design, the proposed scheme performs encryption, encoding and modulation simultaneously. Therefore, we regard the proposed scheme as a joint cryptosystem. The proposed joint cryptosystem withstands all variants of chosen plaintext attacks applied on Rao-Nam like cryptosystems due to its nonlinearity. Moreover, some conditions implying the uniformity of the ciphertexts distribution are introduced through our analysis. Our scheme is efficient and admits small key size. These features are obtained due to several reasons including the quasi-cyclic form of the generator and the parity-check matrices of QC-LDPC lattice-codes, and the simple hardware structure for generating the permutation matrix, the intentional error vector and the nonlinear functions used in our design. The QC-LDPC lattice-codes facilitate high-rate transmission which is suitable for bandlimited AWGN channels. Our simulations indicate that QC-LDPC lattice-codes outperform the error performance of high-order coded modulation schemes based on QAM modulations. Hence, our scheme provides secure, reliable and efficient data transmission in bandlimited AWGN channels.

Low density parity check (LDPC) lattices were the first family of lattices equipped with iterative decoding algorithms. We introduce quasi-cyclic LDPC (QC LDPC) lattices as a special case of LDPC lattices with one binary QC-LDPC code as their underlying code. These lattices are obtained from the Construction A of lattices providing us to encode them efficiently using shift registers. To benefit from an encoder with linear complexity in the lattice dimension, we obtain the generator matrix of these lattices in quasi-cyclic form. We generalize the proposed quasi-cyclic form of the generator matrix for other Construction A lattices, namely the LDA lattices, with a non-binary QC-LDPC code as their underlying code. We provide a low-complexity decoding algorithm of QC LDPC-lattices based on the sum product algorithm. To design lattice codes, QC LDPC-lattices are combined with the nested lattice shaping that uses the Voronoi region of a sublattice for shaping. The shaping gain and the shaping loss of our lattice codes with dimensions 40, 50, and 60 using an optimal quantizer, are presented. The guidelines for applying efficient shaping methods, like hypercube shaping, for QC LDPC-lattices are also given. Consequently, we establish a family of lattice codes that perform practically close to the sphere bound.

Low-density parity-check (LDPC) lattices were the first family of lattices to show efficient decoding in high dimensions. We consider a case of these lattices with one binary LDPC code as an underlying code. We employ encoding and decoding of the LDPC lattices in a cooperative transmission framework. We establish two efficient shaping based on hypercube and Voronoi shaping, to obtain LDPC lattice codes. Then, we propose the implementation of block Markov encoding for one-way and two-way relay networks using LDPC lattice codes. An efficient method is also required for decomposing full-rate codebook into lower rate codebooks. We apply different decomposition schemes for one-way and two-way relay channels, which are the altered versions of the decomposition methods of low density lattice code (LDLC) lattices. Due to the lower complexity of the decoding for LDPC lattices comparing with LDLCs, the complexity of our schemes is significantly lower than the ones proposed for LDLCs. The efficiency of the proposed schemes is presented using simulation results that indicate the outperforming behavior of LDLCs over LDPC lattice codes in the same dimensions. However, having lower decoding complexity enables us to increase the dimension of the lattice to compensate the existing gap between the performance of the LDPC lattice codes and the LDLCs.

Quasi-cyclic (QC) low-density parity-check (LDPC) codes which are known as QC-LDPC codes, have many applications due to their simple encoding implementation by the means of cyclic shift registers. In this paper, we construct QC-LDPC codes from group rings. A group ring is a free module (at the same time a ring) constructed in a natural way from any given ring and any given group. We present a structure based on the elements of a group ring for constructing QC-LDPC codes. Some of the previously addressed methods for constructing QC-LDPC codes based on finite fields are special cases of the proposed construction method. The constructed QC-LDPC codes perform very well over the additive white Gaussian noise channel with iterative decoding in terms of bit-error probability and block-error probability. Simulation results demonstrate that the proposed codes have competitive performance in comparison with the similar existing LDPC codes. Finally, we propose a new encoding method for the proposed group ring-based QC-LDPC codes that can be implemented faster than the current encoding methods. The encoding complexity of the proposed method is analyzed mathematically, and indicates a significate reduction in the required number of operations, even when compared to the available efficient encoding methods that have linear time and space complexities.

LDPC lattices were the first family of lattices which have an efficient decoding algorithm in high dimensions over an AWGN channel. When we consider Construction D' of lattices with one binary LDPC code as its underlying code, 1-level LDPC lattices are obtained. Block fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. In this type of channel, a lattice point is divided into multiple blocks such that fading is constant within a block but changes, independently, across blocks. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. In this paper we construct full diversity 1-level LDPC lattices for block fading channels. We propose a new iterative decoding method for these family of lattices which has complexity that grows linearly in the dimension of lattice.