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Ahlswede and Dueck showed possibility to identify with high probability one out of $M$ messages by transmitting $1/C\log\log M$ bits only, where $C$ is the channel capacity. It is known that this identification can be based on error-correcting codes. We propose an identification procedure based on random codes that achieves channel capacity. Then we show that this procedure can be simplified using pseudo-random generators.

Consider the identification (ID) via channels problem, where a receiver wants to decide whether the transmitted identifier is its identifier, rather than decoding the identifier. This model allows to transmit identifiers whose size scales doubly-exponentially in the blocklength, unlike common transmission (or channel) codes whose size scales exponentially. It suffices to use binary constant-weight codes (CWCs) to achieve the ID capacity. By relating the parameters of a binary CWC to the minimum distance of a code and using higher-order correlation moments, two upper bounds on the binary CWC size are proposed. These bounds are shown to be upper bounds also on the identifier sizes for ID codes constructed by using binary CWCs. We propose two code constructions based on optical orthogonal codes, which are used in optical multiple access schemes, have constant-weight codewords, and satisfy cyclic cross-correlation and auto-correlation constraints. These constructions are modified and concatenated with outer Reed-Solomon codes to propose new binary CWCs optimal for ID. Improvements to the finite-parameter performance of both our and existing code constructions are shown by using outer codes with larger minimum distance vs. blocklength ratios. We also illustrate ID performance regimes for which our ID code constructions perform significantly better than existing constructions., Comment: To appear in the IEEE International Symposium on Information Theory 2021

Quantum synchronizable codes are kinds of quantum error-correcting codes that can not only correct the effects of quantum noise on qubits but also the misalignment in block synchronization. This paper contributes to constructing two classes of quantum synchronizable codes by the cyclotomic classes of order two over $\mathbb{Z}_{2q}$, whose synchronization capabilities can reach the upper bound under certain conditions. Moreover, the quantum synchronizable codes possess good error-correcting capability towards bit errors and phase errors., Comment: arXiv admin note: substantial text overlap with arXiv:2105.03619

This paper investigates the decoding of certain Gabidulin codes that were transmitted over a channel with space-symmetric errors. Space-symmetric errors are additive error matrices that have the property that their column and row spaces are equal. We show that for channels restricted to space-symmetric errors, with high probability errors of rank up to 2(n-k)/3 can be decoded with a Gabidulin code of length n and dimension k, using a weak-self orthogonal basis as code locators.

Two new classes of skew codes over a finite field $\F$ are proposed, called skew convolutional codes and skew trellis codes. These two classes are defined by, respectively, left or right sub-modules over the skew fields of fractions of skew polynomials over $\F$. The skew convolutional codes can be represented as periodic time-varying ordinary convolutional codes. The skew trellis codes are in general nonlinear over $\F$. Every code from both classes has a code trellis and can be decoded by Viterbi or BCJR algorithms., Comment: To appear in IEEE Information Theory Workshop 2020

Interleaved Reed-Solomon codes admit efficient decoding algorithms which correct burst errors far beyond half the minimum distance in the random errors regime, e.g., by computing a common solution to the Key Equation for each Reed-Solomon code, as described by Schmidt et al. If this decoder does not succeed, it may either fail to return a codeword or miscorrect to an incorrect codeword, and good upper bounds on the fraction of error matrices for which these events occur are known. The decoding algorithm immediately applies to interleaved alternant codes as well, i.e., the subfield subcodes of interleaved Reed-Solomon codes, but the fraction of decodable error matrices differs, since the error is now restricted to a subfield. In this paper, we present new general lower and upper bounds on the fraction of error matrices decodable by Schmidt et al.'s decoding algorithm, thereby making it the only decoding algorithm for interleaved alternant codes for which such bounds are known.

A key agreement problem is considered that has a biometric or physical identifier, a terminal for key enrollment, and a terminal for reconstruction. A nested convolutional code design is proposed that performs vector quantization during enrollment and error control during reconstruction. Physical identifiers with small bit error probability illustrate the gains of the design. One variant of the nested convolutional codes improves on the best known key vs. storage rate ratio but it has high complexity. A second variant with lower complexity performs similar to nested polar codes. The results suggest that the choice of code for key agreement with identifiers depends primarily on the complexity constraint., Comment: To appear in ACM Workshop on Information Hiding and Multimedia Security 2020

We consider polar subcodes (PSCs), which are polar codes (PCs) with dynamically-frozen symbols, to increase the minimum distance as compared to corresponding PCs. A randomized nested PSC construction with a low-rate PSC and a high-rate PC, is proposed for list and sequential successive cancellation decoders. This code construction aims to perform lossy compression with side information. Nested PSCs are used in the key agreement problem with physical identifiers. Gains in terms of the secret-key vs. storage rate ratio as compared to nested PCs with the same list size are illustrated to show that nested PSCs significantly improve on nested PCs. The performance of the nested PSCs is shown to improve with larger list sizes, which is not the case for nested PCs considered., Comment: Shorter version to appear in 2020 IEEE International Symposium on Information Theory and Applications. Decoding complexity results are added

Different transforms used in binding a secret key to correlated physical-identifier outputs are compared. Decorrelation efficiency is the metric used to determine transforms that give highly-uncorrelated outputs. Scalar quantizers are applied to transform outputs to extract uniformly distributed bit sequences to which secret keys are bound. A set of transforms that perform well in terms of the decorrelation efficiency is applied to ring oscillator (RO) outputs to improve the uniqueness and reliability of extracted bit sequences, to reduce the hardware area and information leakage about the key and RO outputs, and to maximize the secret-key length. Low-complexity error-correction codes are proposed to illustrate two complete key-binding systems with perfect secrecy, and better secret-key and privacy-leakage rates than existing methods. A reference hardware implementation is also provided to demonstrate that the transform-coding approach occupies a small hardware area., Comment: An extra term in the last page due to the mismatch between the Arxiv compiler and MDPI template is eliminated. No other changes

The two-terminal key agreement problem with biometric or physical identifiers is considered. Two linear code constructions based on Wyner-Ziv coding are developed. The first construction uses random linear codes and achieves all points of the key-leakage-storage regions of the generated-secret and chosen-secret models. The second construction uses nested polar codes for vector quantization during enrollment and for error correction during reconstruction. Simulations show that the nested polar codes achieve privacy-leakage and storage rates that improve on existing code designs. One proposed code achieves a rate tuple that cannot be achieved by existing methods., Comment: To appear in IEEE Transactions on Information Forensics and Security

We show that decoding of $\ell$-Interleaved Gabidulin codes, as well as list-$\ell$ decoding of Mahdavifar--Vardy codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of $\F[x]$ matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form. We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding $\ell$-Interleaved Gabidulin codes. We obtain an algorithm with complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem and is proportional to the code length $n$ in the case of decoding. Further, we show how to perform the interpolation step of list-$\ell$-decoding Mahdavifar--Vardy codes in complexity $O(\ell n^2)$, where $n$ is the number of interpolation constraints., Comment: Accepted for Designs, Codes and Cryptography

A new class of folded subspace codes for noncoherent network coding is presented. The codes can correct insertions and deletions beyond the unique decoding radius for any code rate $R\in[0,1]$. An efficient interpolation-based decoding algorithm for this code construction is given which allows to correct insertions and deletions up to the normalized radius $s(1-((1/h+h)/(h-s+1))R)$, where $h$ is the folding parameter and $s\leq h$ is a decoding parameter. The algorithm serves as a list decoder or as a probabilistic unique decoder that outputs a unique solution with high probability. An upper bound on the average list size of (folded) subspace codes and on the decoding failure probability is derived. A major benefit of the decoding scheme is that it enables probabilistic unique decoding up to the list decoding radius., Comment: 6 pages, 1 figure, accepted for ISIT 2015

For many algebraic codes the main part of decoding can be reduced to a shift register synthesis problem. In this paper we present an approach for solving generalised shift register problems over skew polynomial rings which occur in error and erasure decoding of $\ell$-Interleaved Gabidulin codes. The algorithm is based on module minimisation and has time complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem., Comment: 10 pages, submitted to WCC 2015

Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a non-coherent multi-shot network, where the unknown and time-variant network is used several times. In order to create dependencies between the different shots, particular convolutional codes in rank metric are used. These codes are so-called (partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one. First, distance measures for convolutional codes in rank metric are shown and two constructions of (P)UM codes in rank metric based on the generator matrices of maximum rank distance codes are presented. Second, an efficient error-erasure decoding algorithm for these codes is presented. Its guaranteed decoding radius is derived and its complexity is bounded. Finally, it is shown how to apply these codes for error correction in random linear and affine network coding., Comment: presented in part at Netcod 2012, submitted to IEEE Transactions on Information Theory

We address the problem of increasing the sum rate in a multiple-access system from [1] for small number of users. We suggest an improved signal-code construction in which in case of a small number of users we give more resources to them. For the resulting multiple-access system a lower bound on the relative sum rate is derived. It is shown to be very close to the maximal value of relative sum rate in [1] even for small number of users. The bound is obtained for the case of decoding by exhaustive search. We also suggest reduced-complexity decoding and compare the maximal number of users in this case and in case of decoding by exhaustive search., Comment: 5 pages, 4 figures, submitted to IEEE ISIT 2013

Traditionally, multi-trial error/erasure decoding of Reed-Solomon (RS) codes is based on Bounded Minimum Distance (BMD) decoders with an erasure option. Such decoders have error/erasure tradeoff factor L=2, which means that an error is twice as expensive as an erasure in terms of the code's minimum distance. The Guruswami-Sudan (GS) list decoder can be considered as state of the art in algebraic decoding of RS codes. Besides an erasure option, it allows to adjust L to values in the range 1=1 times. We show that BMD decoders with z_BMD decoding trials can result in lower residual codeword error probability than GS decoders with z_GS trials, if z_BMD is only slightly larger than z_GS. This is of practical interest since BMD decoders generally have lower computational complexity than GS decoders., Comment: Accepted for the 2011 IEEE International Symposium on Information Theory, St. Petersburg, Russia, July 31 - August 05, 2011. 5 pages, 2 figures

Algebraic decoding algorithms are commonly applied for the decoding of Reed-Solomon codes. Their main advantages are low computational complexity and predictable decoding capabilities. Many algorithms can be extended for correction of both errors and erasures. This enables the decoder to exploit binary quantized reliability information obtained from the transmission channel: Received symbols with high reliability are forwarded to the decoding algorithm while symbols with low reliability are erased. In this paper we investigate adaptive single-trial error/erasure decoding of Reed-Solomon codes, i.e. we derive an adaptive erasing strategy which minimizes the residual codeword error probability after decoding. Our result is applicable to any error/erasure decoding algorithm as long as its decoding capabilities can be expressed by a decoder capability function. Examples are Bounded Minimum Distance decoding with the Berlekamp-Massey- or the Sugiyama algorithms and the Guruswami-Sudan list decoder., Comment: Accepted for the 2011 Canadian Workshop on Information Theory, Kelowna, BC, Canada, May 17 - 20, 2011. 5 pages, 4 figures

(Partial) Unit Memory ((P)UM) codes provide a powerful possibility to construct convolutional codes based on block codes in order to achieve a high decoding performance. In this contribution, a construction based on Gabidulin codes is considered. This construction requires a modified rank metric, the so-called sum rank metric. For the sum rank metric, the free rank distance, the extended row rank distance and its slope are defined analogous to the extended row distance in Hamming metric. Upper bounds for the free rank distance and the slope of (P)UM codes in the sum rank metric are derived and an explicit construction of (P)UM codes based on Gabidulin codes is given, achieving the upper bound for the free rank distance.

We present and prove the correctness of an efficient algorithm that provides a basis for all solutions of a key equation in order to decode Gabidulin (G-) codes up to a given radius tau. This algorithm is based on a symbolic equivalent of the Euclidean Algorithm (EA) and can be applied for decoding of G-codes beyond half the minimum rank distance. If the key equation has a unique solution, our algorithm reduces to Gabidulin's decoding algorithm up to half the minimum distance. If the solution is not unique, we provide a basis for all solutions of the key equation. Our algorithm has time complexity O(tau^2) and is a generalization of the modified EA by Bossert and Bezzateev for Reed-Solomon codes., Comment: accepted for ISIT 2010, Austin, TX, USA

We investigate adaptive single-trial error/erasure decoding of binary codes whose decoder is able to correct e errors and t erasures if le+t<=d-1. Thereby, d is the minimum Hamming distance of the code and 1

We investigate threshold-based multi-trial decoding of concatenated codes with an inner Maximum-Likelihood decoder and an outer error/erasure (L+1)/L-extended Bounded Distance decoder, i.e. a decoder which corrects e errors and t erasures if e(L+1)/L + t <= d - 1, where d is the minimum distance of the outer code and L is a positive integer. This is a generalization of Forney's GMD decoding, which was considered only for L = 1, i.e. outer Bounded Minimum Distance decoding. One important example for (L+1)/L-extended Bounded Distance decoders is decoding of L-Interleaved Reed-Solomon codes. Our main contribution is a threshold location formula, which allows to optimally erase unreliable inner decoding results, for a given number of decoding trials and parameter L. Thereby, the term optimal means that the residual codeword error probability of the concatenated code is minimized. We give an estimation of this probability for any number of decoding trials., Comment: Accepted for the 2010 IEEE International Symposium on Information Theory, Austin, TX, USA, June 13 - 18, 2010. 5 pages, 2 figures

We consider the Additive White Gaussian Noise channel with Binary Phase Shift Keying modulation. Our aim is to enable an algebraic hard decision Bounded Minimum Distance decoder for a binary block code to exploit soft information obtained from the demodulator. This idea goes back to Forney and is based on treating received symbols with low reliability as erasures. This erasing at the decoder is done using a threshold, each received symbol with reliability falling below the threshold is erased. Depending on the target overall complexity of the decoder this pseudo-soft decision decoding can be extended from one threshold T to z>1 thresholds T_1<...1 thresholds case, starting with a geometric interpretation of the optimal threshold location problem and using an approach from Zyablov., Comment: Accepted for the XII International Symposium on Problems of Redundancy in Information and Control Systems, St. Petersburg, Russia, May 26 - 30, 2009. 5 pages, 3 figures

Generalized Concatenated codes are a code construction consisting of a number of outer codes whose code symbols are protected by an inner code. As outer codes, we assume the most frequently used Reed-Solomon codes; as inner code, we assume some linear block code which can be decoded up to half its minimum distance. Decoding up to half the minimum distance of Generalized Concatenated codes is classically achieved by the Blokh-Zyablov-Dumer algorithm, which iteratively decodes by first using the inner decoder to get an estimate of the outer code words and then using an outer error/erasure decoder with a varying number of erasures determined by a set of pre-calculated thresholds. In this paper, a modified version of the Blokh-Zyablov-Dumer algorithm is proposed, which exploits the fact that a number of outer Reed-Solomon codes with average minimum distance d can be grouped into one single Interleaved Reed-Solomon code which can be decoded beyond d/2. This allows to skip a number of decoding iterations on the one hand and to reduce the complexity of each decoding iteration significantly - while maintaining the decoding performance - on the other., Comment: Proceedings of the 2008 IEEE International Symposium on Information Theory, Toronto, ON, Canada, July 6 - 11, 2008. 5 pages, 2 figures

For a certain class of functions, the distribution of the function values can be calculated in the trellis or a sub-trellis. The forward/backward recursion known from the BCJR algorithm is generalized to compute the moments of these distributions. In analogy to the symbol probabilities, by introducing a constraint at a certain depth in the trellis we obtain symbol moments. These moments are required for an efficient implementation of the discriminated belief propagation algorithm in [2], and can furthermore be utilized to compute conditional entropies in the trellis. The moment computation algorithm has the same asymptotic complexity as the BCJR algorithm. It is applicable to any commutative semi-ring, thus actually providing a generalization of the Viterbi algorithm., Comment: 9 pages, 4 figures

In this paper, a new approach for decoding low-rate Reed-Solomon codes beyond half the minimum distance is considered and analyzed. Unlike the Sudan algorithm published in 1997, this new approach is based on multi-sequence shift-register synthesis, which makes it easy to understand and simple to implement. The computational complexity of this shift-register based algorithm is of the same order as the complexity of the well-known Berlekamp-Massey algorithm. Moreover, the error correcting radius coincides with the error correcting radius of the original Sudan algorithm, and the practical decoding performance observed on a q-ary symmetric channel (QSC) is virtually identical to the decoding performance of the Sudan algorithm. Bounds for the failure and error probability as well as for the QSC decoding performance of the new algorithm are derived, and the performance is illustrated by means of examples., Comment: 14 pages, 5 figures

Interleaved Reed-Solomon codes are applied in numerous data processing, data transmission, and data storage systems. They are generated by interleaving several codewords of ordinary Reed-Solomon codes. Usually, these codewords are decoded independently by classical algebraic decoding methods. However, by collaborative algebraic decoding approaches, such interleaved schemes allow the correction of error patterns beyond half the minimum distance, provided that the errors in the received signal occur in bursts. In this work, collaborative decoding of interleaved Reed-Solomon codes by multi-sequence shift-register synthesis is considered and analyzed. Based on the framework of interleaved Reed-Solomon codes, concatenated code designs are investigated, which are obtained by interleaving several Reed-Solomon codes, and concatenating them with an inner block code., Comment: 20 pages, 7 figures

The problem of finding the shortest linear shift-register capable of generating t finite length sequences over some field F is considered. A similar problem was already addressed by Feng and Tzeng. They presented an iterative algorithm for solving this multi-sequence shift-register synthesis problem, which can be considered as generalization of the well known Berlekamp-Massey algorithm. The Feng-Tzeng algorithm works indeed, if all t sequences have the same length. This paper focuses on multi-sequence shift-register synthesis for generating sequences of varying length. It is exposed, that the Feng-Tzeng algorithm does not always give the correct solution in this case. A modified algorithm is proposed and formally proved, which overcomes this problem.