Residue Codes Research Articles

Infinite series of 3‐designs in the extended quadratic residue code

AbstractIn this paper, we show an infinite series of 3‐designs in the extended quadratic residue codes over for a prime .

Eight Error Correction for (57,29,17) Quadratic Residue Code Over Binary Field

Eight Error Correction for (57,29,17) Quadratic Residue Code Over Binary Field

Neural layered min-sum decoders for cyclic codes

This paper proposes a low-complexity neural network decoder based on the layered min-sum algorithm to decode cyclic codes. By generalizing the layered min-sum algorithm to its neural network counterpart, the number of network weights decreases while retaining a good error correction performance. The Bose–Chaudhuri–Hocquenghem (BCH) codes, quadratic residue (QR) codes, and punctured Reed–Muller (RM) codes are selected as three exemplary binary cyclic codes. Simulation results show that the proposed neural decoder achieves superior performance with less computational complexity compared with the state-of-the-art neural network decoder. Further, a neural decoder incorporating the modified random redundant decoding (mRRD) algorithm is investigated to approach the performance of maximum-likelihood decoding for some short codes.

Exceptional designs in some extended quadratic residue codes

AbstractIn the present paper, we give proofs of the existence of a 3‐design in the extended ternary quadratic residue code of length 14 and the extended quaternary quadratic residue code of length 18.

Improved Spacecraft Authentication Method for Satellite Internet System Using Residue Codes

Low-orbit satellite internet (LOSI) expands the scope of the Industrial Internet of Things (IIoT) in the oil and gas industry (OGI) to include areas of the Far North. However, due to the large length of the communication channel, the number of threats and attacks increases. A special place among them is occupied by relay spoofing interference. In this case, an intruder satellite intercepts the control signal coming from the satellite (SC), delays it, and then imposes it on the receiver located on the unattended OGI object. This can lead to a disruption of the facility and even cause an environmental disaster. To prevent a spoofing attack, a satellite authentication method has been developed that uses a zero-knowledge authentication protocol (ZKAP). These protocols have high cryptographic strength without the use of encryption. However, they have a significant drawback. This is their low authentication speed, which is caused by calculations over a large module Q (128 bits or more). It is possible to reduce the time of determining the status of an SC by switching to parallel computing. To solve this problem, the paper proposes to use residue codes (RC). Addition, subtraction, and multiplication operations are performed in parallel in RC. Thus, a correct choice of a set of modules of RC allows for providing an operating range of calculations not less than the number Q. Therefore, the development of a spacecraft authentication method for the satellite internet system using RC that allows for reducing the authentication time is an urgent task.

Revisiting Residue Codes for Modern Memories

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">This paper shows how residue codes, traditionally used for compute rather than storage error correction, can be applied to memories with surprising results. We show that adapting residue codes to modern memory systems offers a level of error correction comparable to traditional schemes such as Reed-Solomon but with fewer bits of storage. For instance, our adaptation of residue codes –</i> MUSE ECC <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">– can offer ChipKill protection using approximately 30% fewer bits. We use the storage gains to hold metadata needed for emerging security functionality such as memory tagging or to provide better detection capabilities against Rowhammer attacks. In a system with memory tagging and</i> MUSE <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">, we achieve a 12% reduction in memory bandwidth utilization with the same error correction level as a traditional ECC baseline and without a noticeable performance loss. Thus, our work demonstrates a new, flexible primitive for co-designing reliability with security and performance.</i>

Efficient Decoding Algorithm for Binary Quadratic Residue Codes Using Reduced Permutation Sets

The Quadratic Residue (QR) codes have a rich mathematic structure. Unfortunately, their Algebraic Decoding (AD) is not generalizable for all QR codes. In this study, an efficient hard decoding algorithm is proposed to generalize the decoding of the binary systematic Quadratic Residue (QR) codes. The proposed decoder corrects t erroneous bits or less, in the received word, based on a reduced set of permutations derived from the large automorphism group of QR codes. This set of permutations is applied to the received word to move the error positions and trap all of them in redundancy. Then, to evaluate the proposed method, we applied it to many binary QR codes of moderate code length starting with 17 until 113 with reducible and irreducible generator polynomials. The proposed decoder was validated by inserting all possible error patterns, that have t or less erroneous positions, as input of the proposed decoder and the output is always a correct codeword. The complexity study, in terms of the number of operations used, reveals that the light permutation decoding LPD algorithm significantly decreases decoding complexity without performance loss. So, it is qualified to be a good competitor to decode QR codes with lower lengths but is the best for QR codes with higher lengths.

A Simple Neural-Network-Based Decoder for Short Binary Linear Block Codes

The conventional soft decision decoding (SDD) methods require various hard decision decoders (HDDs) based on different codes or re-manipulate the generator matrix by the complicated Gaussian elimination technique according to the bit reliability. This paper presents a general multi-class neural network (NN)-based decoder for the short linear block codes, where no HDD and Gaussian elimination are required once the NN is constructed. This network architecture performs multi-classification to select the messages with high occurrence probabilities and chooses the best codeword on a maximum likelihood basis. Simulation results show that the developed approach outperforms the existing deep neural network (DNN)-based decoders in terms of decoding time and bit error rate (BER). The error-correcting performance is also superior to the conventional Chase-II algorithm and is close to the ordered statistics decoding (OSD) in most cases. For Bose–Chaudhuri–Hocquenghem (BCH) codes, the SNR is improved by 1dB to 4dB as the BER is 10−4. For the (23, 12) quadratic residue (QR) code, the SNR is improved by 2dB when the BER is 10−3. The developed NN-based decoder is quite general and applicable to various short linear block codes with good BER performance.

Schematic 4-designs

We study 4-designs with three intersection numbers. By the Cameron-Delsarte theorem, the blocks form a symmetric three-class association scheme. This imposes strong restrictions on the parameters of such designs. We calculate the eigenvalues of the association scheme from the design parameters and determine all admissible parameters with at most 1000 points. An infinite family of admissible parameters is discovered. Designs with small admissible parameters exist and are related to the quadratic residue codes.

Five errors correcting (73, 61, 12) quadratic residue code over ternary field

Five errors correcting (73, 61, 12) quadratic residue code over ternary field

An Efficient CRT-Base Power-of-Two Scaling in Minimally Redundant Residue Number System.

In this paper, we consider one of the key problems in modular arithmetic. It is known that scaling in the residue number system (RNS) is a rather complicated non-modular procedure, which requires expensive and complex operations at each iteration. Hence, it is time consuming and needs too much hardware for implementation. We propose a novel approach to power-of-two scaling based on the Chinese Remainder Theorem (CRT) and rank form of the number representation in RNS. By using minimal redundancy of residue code, we optimize and speed up the rank calculation and parity determination of divisible integers in each iteration. The proposed enhancements make the power-of-two scaling simpler and faster than the currently known methods. After calculating the rank of the initial number, each iteration of modular scaling by two is performed in one modular clock cycle. The computational complexity of the proposed method of scaling by a constant Sl=2l associated with both required modular addition operations and lookup tables is estimeted as k and 2k+1, respectively, where k equals the number of primary non-redundant RNS moduli. The time complexity is log2k+l modular clock cycles.

Efficient Decoder for Turbo Product Codes Based on Quadratic Residue Codes

In this letter, we study turbo product codes with quadratic residue codes (called QR-TPCs) as the component codes. We propose an efficient decoder based on Chase-II algorithm with two convergence conditions for the iterative decoding of QR-TPCs. For each row and column, the Chase-II decoder will stop immediately when one of the conditions is met. The simulation results show that the proposed algorithm has a lower computational complexity compared with existing decoding methods. Moreover, a comparison with 5G low-density parity-check codes shows that the proposed turbo product codes have better performance for short code lengths.

Decoding Quadratic Residue Codes Using Deep Neural Networks

In this paper, a low-complexity decoder based on a neural network is proposed to decode binary quadratic residue (QR) codes. The proposed decoder is based on the neural min-sum algorithm and the modified random redundant decoder (mRRD) algorithm. This new method has the same asymptotic time complexity as the min-sum algorithm, which is much lower than the difference on syndromes (DS) algorithm. Simulation results show that the proposed algorithm achieves a gain of more than 0.4 dB when compared to the DS algorithm. Furthermore, a simplified approach based on trapping sets is applied to reduce the complexity of the mRRD. This simplification leads to a slight loss in error performance and a reduction in implementation complexity.

M-adic residue codes over F_q[v]/(v^s-v) and their applications to quantum codes

Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The $m$-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the $m$-adic residue codes over the quotient ring $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We determine the idempotent generators of the $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$. We obtain some parameters of optimal $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ with respect to Griesmer bound for rings. Furthermore, we derive a condition for $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing $m$-adic residue codes over $\frac{{{\mathbb{F}_q}\left[ v \right]}}{{\left\langle {{v^s} - v} \right\rangle }}$ and give some examples to illustrate our findings.

An Efficient Parallel Reverse Conversion of Residue Code to Mixed-Radix Representation Based on the Chinese Remainder Theorem.

In this paper, we deal with the critical problems in residue arithmetic. The reverse conversion from a Residue Number System (RNS) to positional notation is a main non-modular operation, and it constitutes a basis of other non-modular procedures used to implement various computational algorithms. We present a novel approach to the parallel reverse conversion from the residue code into a weighted number representation in the Mixed-Radix System (MRS). In our proposed method, the calculation of mixed-radix digits reduces to a parallel summation of the small word-length residues in the independent modular channels corresponding to the primary RNS moduli. The computational complexity of the developed method concerning both required modular addition operations and one-input lookup tables is estimated as , where k equals the number of used moduli. The time complexity is modular clock cycles. In pipeline mode, the throughput rate of the proposed algorithm is one reverse conversion in one modular clock cycle.

On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs

It is proved that a code L(q) which is monomially equivalent to the Pless symmetry code C(q) of length $$2q+2$$ contains the (0,1)-incidence matrix of a Hadamard 3- $$(2q+2,q+1,(q-1)/2)$$ design D(q) associated with a Paley–Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley–Hadamard matrix of type I. If $$q=5, 11, 17, 23$$ , then the full permutation automorphism group of L(q) coincides with the full automorphism group of D(q), and a similar result holds for the ternary extended quadratic residue codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by codewords of the Pless symmetry code C(17) are enumerated and classified up to equivalence. There are two equivalence classes of such matrices: the Paley–Hadamard matrix H of type I with a full automorphism group of order 19584, and a second regular Hadamard matrix $$H'$$ such that the symmetric 2-(36, 15, 6) design D associated with $$H'$$ has trivial full automorphism group, and the incidence matrix of D spans a ternary code equivalent to C(17).

Quadratic residue codes, rank three groups and PBIBDs

The automorphism group of the Zetterberg code Z of length 17 (also a quadratic residue code) is a rank three group whose orbits on the coordinate pairs determine two strongly regular graphs equivalent to the Paley graph attached to the prime 17. As a consequence, codewords of a given weight of Z are the characteristic vectors of the blocks of a PBIBD with two associate classes of cyclic type. More generally, this construction of PBIBDs is extended to quadratic residue codes of length \(\equiv 1 \pmod {8},\) to the adjacency codes of triangular and lattice graphs, and to the adjacency codes of various rank three graphs. A remarkable fact is the existence of 2-designs held by the quadratic residue code of length 41 for code weights 9 and 10.

The extended binary quadratic residue code of length 42 holds a 3‐design

AbstractThe codewords of weight 10 of the [42, 21, 10] extended binary quadratic residue code are shown to hold a design of parameters . Its automorphism group is isomorphic to . Its existence can be explained neither by a transitivity argument, nor by the Assmus–Mattson theorem.

A residue codes and berger codes combined coding self-check method for embryonic electronic cell

A residue codes and berger codes combined coding self-check method for embryonic electronic cell

Computationally Efficient Approach to Implementation of the Chinese Remainder Theorem Algorithm in Minimally Redundant Residue Number System

In this paper, we deal with the critical problem of performing non-modular operations in the Residue Number System (RNS). The Chinese Remainder Theorem (CRT) is widely used in many modern computer applications. Throughout the article, an efficient approach for implementing the CRT algorithm is described. The structure of the rank of an RNS number, a principal positional characteristic of the residue code, is investigated. It is shown that the rank of a number can be represented by a sum of an inexact rank and a two-valued correction to it. We propose a new variant of minimally redundant RNS, which provides low computational complexity for the rank calculation, and its effectiveness analyzed concerning conventional non-redundant RNS. Owing to the extension of the residue code, by adding the excess residue modulo 2, the complexity of the rank calculation goes down from Oleft (k^{2}right ) to Oleft (kright ) with respect to required modular addition operations and lookup tables, where k equals the number of non-redundant RNS moduli.