Class Of Error-correcting Codes Research Articles

Heights of error-correcting codes

In this work, we investigate the evaluation of odd polynomials P defined on a finite field on the class of error-correcting codes C. We exploit the correspondence between codes and Tanner graphs. Thus, we formally define P(C), a polynomial code. Then the new notion of height of a code emerges, whose properties are studied. We extended the lower bound of Tanner on the minimum distance of a code to the case of a polynomial code, by using spectral graph theory. Computer algebra software enable us to give numerical results to illustrate the theory of polynomial codes for various classes of error-correcting codes.

Elliptic genera from classical error-correcting codes

We consider chiral fermionic conformal field theories constructed from classical error-correcting codes and provide a systematic way of computing their elliptic genera. We exploit the U(1) current of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal algebra to obtain the U(1)-graded partition function that is invariant under the modular transformation and the spectral flow. We demonstrate our method by constructing extremal N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 elliptic genera from classical codes for relatively small central charges. Also, we give near-extremal elliptic genera and decompose them into N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal characters.

BER performance analysis of polar-coded FBMC/OQAM in the presence of AWGN and Nakagami-m fading channel

<abstract> <p>Offset quadrature amplitude modulation–based filter bank multicarrier (FBMC-OQAM) method is a promising technology for future wireless communication systems. It offers several advantages over traditional orthogonal frequency-division multiplexing (OFDM) modulation, including higher spectral efficiency, lower out-of-band emission, and improved robustness to time-frequency selective channels. Polar codes, a new class of error-correcting codes, have received much attention recently due to their ability to achieve the Shannon limit with practical decoding complexity. This paper analyzed and investigated the error rate performance of polar-coded FBMC-OQAM systems. Our results show that applying polar codes to FBMC-OQAM systems significantly improves the error rate. In addition, we found that employing random code interleavers can yield additional coding gains of up to 0.75 dB in additive white Gaussian noise (AWGN) and 2 dB in Nakagami-m fading channels. Our findings suggest that polar-coded FBMC-OQAM is a promising combination for future wireless communication systems. We also compared turbo-coded FBMC-OQAM for short code lengths, and our simulations showed that polar codes exhibit comparable error-correcting capabilities. These results will be of interest to researchers and engineers working on the advancement of future wireless communication systems.</p> </abstract>

One-time memory from isolated Majorana islands

A one-time memory (OTM) stores two bits such that either bit can be retrieved, but not both. One-time memories have been shown to allow for secure classical and quantum computations. We propose a physical construction of this mathematical cryptographic primitive, based on isolated Majorana islands. Using classical error-correcting codes, obtained from a quantum code, the scheme can be made robust to errors. Further, the one bit out of two bit OTM construction generalizes to 1-out-of- and -out-of- OTMs.

Classical codes and chiral CFTs at higher genus

Higher genus modular invariance of two-dimensional conformal field theories (CFTs) is a largely unexplored area. In this paper, we derive explicit expressions for the higher genus partition functions of a specific class of CFTs: code CFTs, which are constructed using classical error-correcting codes. In this setting, the Sp(2g, ℤ) modular transformations of genus g Riemann surfaces can be recast as a simple set of linear maps acting on 2g polynomial variables, which comprise an object called the code enumerator polynomial. The CFT partition function is directly related to the enumerator polynomial, meaning that solutions of the linear constraints from modular invariance immediately give a set of seemingly consistent partition functions at a given genus. We then find that higher genus constraints, plus consistency under degeneration limits of the Riemann surface, greatly reduces the number of possible code CFTs. This work provides a step towards a full understanding of the constraints from higher genus modular invariance on 2d CFTs.

Decoding algorithms of monotone codes and azinv codes and their unified view

This paper investigates linear-time decoding algorithms for two classes of error-correcting codes. One of the classes is monotone codes which are known as single deletion error-correcting codes, although they are not known to be single substitution error-correcting codes. The other is azinv codes which are known as single balanced adjacent deletion error-correcting codes, although they are not known to be single balanced adjacent substitution error-correcting codes. As a result, this paper proposes generalizations of Levenshtein’s decoding algorithm for Levenshtein’s single deletion or single substitution error-correcting codes. This paper points out that it is possible to unify our new two decoding algorithms. Moreover, we provide Python implementations of these algorithms and the graphs of their computational costs at https://github.com/Hokuto496/Decoding_Algorithms_of_monotone_codes_and_azinv_codes.

Performance analysis of Min-Sum based LDPC decoder architecture for 5G new radio standards

A low-density parity-check code (LDPC) is one of the classical error-correcting code for message transmission over noisy channel. It meets desired Shannon limit performance and is adopted as a data channel for the 5th generation new radio (5G NR) standard. This paper investigates the LDPC decoder by applying minimum-sum (MS) algorithm on the base matrix (BG1) of 5G standards. The performance of the decoder such as bit-error-rate (BER) and frame-error-rate (FER) has been improved when the number of iterations increased for larger code word. The variable node and check node architecture based on the MS algorithm with pipelining stage has been implemented in the field programmable gate array (FPGA). Based on the implementation results, it is observed that the LDPC decoder shows better performance in throughput and hardware resource usage (HUE) with some additional look-up-tables (LUT) and flip-flops (FF).

Error Probability Mitigation in Quantum Reading Using Classical Codes.

A general framework describing the statistical discrimination of an ensemble of quantum channels is given by the name quantum reading. Several tools can be applied in quantum reading to reduce the error probability in distinguishing the ensemble of channels. Classical and quantum codes can be envisioned for this goal. The aim of this paper is to present a simple but fruitful protocol for this task using classical error-correcting codes. Three families of codes are considered: Reed–Solomon codes, BCH codes, and Reed–Muller codes. In conjunction with the use of codes, we also analyze the role of the receiver. In particular, heterodyne and Dolinar receivers are taken into consideration. The encoding and measurement schemes are connected by the probing step. As probes, we consider coherent states. In such a simple manner, interesting results are obtained. As we show, there is a threshold below which using codes surpass optimal and sophisticated schemes for any fixed rate and code. BCH codes in conjunction with Dolinar receiver turn out to be the optimal strategy for error mitigation in quantum reading.

Classifying toric surface codes of dimension 7

Toric surface codes are a class of error-correcting codes coming from a lattice polytope defining a two-dimensional toric variety. Previous authors have mostly completed classifications of these toric surface codes with dimension up to $k = 7.$ In this note, we correct an error in the classification of the $k=7$ case started in \cite{HLYZZ}, and disprove one of their conjectures.

Quantum stabilizer codes, lattices, and CFTs

There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E8 theory, which is based on the root lattice of E8 understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples.

О ГРАНИЦАХ МОЩНОСТИ ЗЛОУМЫШЛЕННИКОВ ДЛЯ ИДЕНТИФИЦИРУЮЩИХ АЛГЕБРОГЕОМЕТРИЧЕСКИХ КОДОВ НА СПЕЦИАЛЬНЫХ КРИВЫХ

Broadcast encryption is a data distribution protocol which can prevent malefactor parties from unauthorized accessing or copying the distributed data. It is widely used in distributed storage and network data protection schemes. To block the socalled coalition attacks on the protocol, classes of error-correcting codes with special properties are used, namely c-FP and c-TA properties. We study the problem of evaluating the lower and the upper boundaries on coalition power, within which the algebraic geometry codes possess these properties. Earlier, these boundaries were calculated for single-point algebraic-geometric codes on curves of the general form. Now, we clarified these boundaries for single-point codes on curves of a special form; in particular, for codes on curves on which there are many equivalence classes after factorization by equality of the corresponding points coordinates relation.

Magic state distillation with the ternary Golay code.

The ternary Golay code-one of the first and most beautiful classical error-correcting codes discovered-naturally gives rise to an 11-qutrit quantum error correcting code. We apply this code to magic state distillation, a leading approach to fault-tolerant quantum computing. We find that the 11-qutrit Golay code can distil the 'most magic' qutrit state-an eigenstate of the qutrit Fourier transform known as the strange state-with cubic error suppression and a remarkably high threshold. It also distils the 'second-most magic' qutrit state, the Norell state, with quadratic error suppression and an equally high threshold to depolarizing noise.

On the Properties of Algebraic Geometric Codes as Copy Protection Codes

Traceability schemes which are applied to the broadcast encryption can prevent unauthorized parties from accessing the distributed data. In a traceability scheme a distributor broadcasts the encrypted data and gives each authorized user unique key and identifying word from selected error-correcting code for decrypting. The following attack is possible in these schemes: groups of c malicious users are joining into coalitions and gaining illegal access to the data by combining their keys and identifying codewords to obtain pirate key and codeword. To prevent this attacks, classes of error-correcting codes with special c-FP and c-TA properties are used. In particular, c -FP codes are codes that make direct compromise of scrupulous users impossible and c -TA codes are codes that make it possible to identify one of the a‹ackers. We are considering the problem of evaluating the lower and the upper boundaries on c, within which the L-construction algebraic geometric codes have the corresponding properties. In the case of codes on an arbitrary curve the lower bound for the c-TA property was obtained earlier; in this paper, the lower bound for the c-FP property was constructed. In the case of curves with one infinite point, the upper bounds for the value of c are obtained for both c-FP and c-TA properties. During our work, we have proved an auxiliary lemma and the proof contains an explicit way to build a coalition and a pirate identifying vector. Methods and principles presented in the lemma can be important for analyzing broadcast encryption schemes robustness. Also, the c-FP and c-TA boundaries monotonicity by subcodes are proved.

Quadratic and symmetric bilinear forms over finite fields and their association schemes

Let 𝒬(m,q) and 𝒮(m,q) be the sets of quadratic forms and symmetric bilinear forms on an m-dimensional vector space over 𝔽 q , respectively. The orbits of 𝒬(m,q) and 𝒮(m,q) under a natural group action induce two translation association schemes, which are known to be dual to each other. We give explicit expressions for the eigenvalues of these association schemes in terms of linear combinations of generalised Krawtchouk polynomials, generalising earlier results for odd q to the more difficult case when q is even. We then study d-codes in these schemes, namely subsets X of 𝒬(m,q) or 𝒮(m,q) with the property that, for all distinct A,B∈X, the rank of A-B is at least d. We prove tight bounds on the size of d-codes and show that, when these bounds hold with equality, the inner distributions of the subsets are often uniquely determined by their parameters. We also discuss connections to classical error-correcting codes and show how the Hamming distance distribution of large classes of codes over 𝔽 q can be determined from the results of this paper.

Non-split Toric Codes

We introduce a new wide class of error-correcting codes, called non-split toric codes. These codes are a natural generalization of toric codes where non-split algebraic tori are taken instead of usual (i.e., split) ones. The main advantage of the new codes is their cyclicity; hence, they can possibly be decoded quite fast. Many classical codes, such as (doubly-extended) Reed-Solomon and (projective) Reed-Muller codes, are contained (up to equivalence) in the new class. Our codes are explicitly described in terms of algebraic and toric geometries over finite fields; therefore, they can easily be constructed in practice. Finally, we obtain new cyclic reversible codes, namely non-split toric codes on the del Pezzo surface of degree 6 and Picard number 1. We also compute their parameters, which prove to attain current lower bounds at least for small finite fields.

Algorithm for the generation of complement-free sets

We introduce an algorithm for the generation of complement-free sets of binary m-tuples, where m is even. We also provide an implementation for this algorithm for m = 12. Such complement-free sets are needed for the generation of a new class of error-correcting codes, which were introduced by Hannusch and Lakatos. These codes build the fundamental improvement in the cryptographic system of Domosi, Hannusch and Horvath. Therefore the generation of complement-free sets will be important for cryptographic applications. In the end of the paper we give some interesting facts about complement-free sets as combinatorial objects.

A construction of q-ary linear codes with irreducible cyclic codes

Linear codes are an important class of error-correcting codes and widely used in secret sharing schemes, combinational designs, authentication codes and so on. The objective of this paper is to construct q-ary linear codes with good parameters from irreducible cyclic codes. Several classes of linear codes with a few weights including infinite families of distance-optimal ones are derived. The nonzero codewords of some of the codes in this paper have minimal support for inclusion and such codes can be used to construct secret sharing schemes with interesting access structures.

Codes in Dihedral Group Algebra

Robert McEliece developed an asymmetric encryption algorithm based on the use of binary Goppa codes in 1978 and no effective key attacks has been described yet. Variants of this cryptosystem are known due to the use of different codes types, but most of them were proven to be less secure. Code cryptosystems are considered an alternate to number-theoretical ones in connection with the development of quantum computing. So, the new classes of error-correcting codes are required for building new resistant code cryptosystems. Non-commutative codes, which simply are ideals of finite non-commutative group algebras, are an option. The Artin–Wedderburn theorem implies that a group algebra is isomorphic to a finite direct sum of matrix algebras, when the order of the group and the field characteristics are relatively prime. This theorem is important to study the structure of a non-commutative code, but it gives no information about summands and the isomorphism. In case of a dihedral group these summands and the isomorphism were found by F. E. Brochero Martinez. The purpose of the paper is to study codes in dihedral group algebras as and when the order of a group and a field characteristics are relatively prime. Using the result of F. E. Brochero Martinez, we consider a structure of all dihedral codes in this case and the codes induced by cyclic subgroup codes.

Efficient preparation of large-block-code ancilla states for fault-tolerant quantum computation

Fault-tolerant quantum computation (FTQC) schemes that use multi-qubit large block codes can potentially reduce the resource overhead to a great extent. A major obstacle is the requirement of a large number of clean ancilla states of different types without correlated errors inside each block. These ancilla states are usually logical stabilizer states of the data code blocks, which are generally difficult to prepare if the code size is large. Previously we have proposed an ancilla distillation protocol for Calderbank-Shor-Steane (CSS) codes by classical error-correcting codes. It was assumed that the quantum gates in the distillation circuit were perfect; however, in reality, noisy quantum gates may introduce correlated errors that are not treatable by the protocol. In this paper, we show that additional postselection by another classical error-detecting code can be applied to remove almost all correlated errors. Consequently, the revised protocol is fully fault-tolerant and capable of preparing a large set of stabilizer states sufficient for FTQC using large block codes. At the same time, the yield rate can be boosted from $O(t^{-2})$ to $O(1)$ in practice for an $[[n,k,d=2t+1]]$ CSS code. Ancilla preparation for the $[[23,1,7]]$ quantum Golay code is numerically studied in detail through Monte Carlo simulation. The results support the validity of the protocol when the gate failure rate is reasonably low. To the best of our knowledge, this approach is the first attempt to prepare general large block stabilizer states free of correlated errors for FTQC in a fault-tolerant and efficient manner.

Belief propagation decoding of quantum channels by passing quantum messages

The belief propagation (BP) algorithm is a powerful tool in a wide range of disciplines from statistical physics to machine learning to computational biology, and is ubiquitous in decoding classical error-correcting codes. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding of the channel, in some cases even up to the Shannon capacity. Here we construct the first BP algorithm which passes quantum messages on the factor graph and is capable of decoding the classical–quantum channel with pure state outputs. This gives explicit decoding circuits whose number of gates is quadratic in the code length. We also show that this decoder can be modified to work with polar codes for the pure state channel and as part of a decoder for transmitting quantum information over the amplitude damping channel. These represent the first explicit capacity-achieving decoders for non-Pauli channels.