In this paper we consider combinatorial secure codes in traitor tracing for protecting copyright of multimedia content. First, we introduce a new notion of secure codes with list decoding (SCLDs) for collusion-resistant multimedia fingerprinting, which includes many existing types of fingerprinting codes as special cases. Next, we build efficient identifying algorithms for SCLDs with complete traceability and establish bounds on its largest possible code rate. In comparison with the existing fingerprinting codes, it is shown that SCLDs have not only much more efficient traceability than separable codes but also a much larger code rate than frameproof codes. As a byproduct, new bounds on the largest code rate of binary separable codes are established as well. Furthermore, a two-stage dynamic traitor tracing framework is proposed for multimedia fingerprinting in the dynamic scenario, which could not only efficiently achieve the complete traceability but also provide a much larger capacity than the static scenario.
We propose a new concept of secure list decoding. While the conventional list decoding requires that the list contains the transmitted message, secure list decoding requires the following additional security conditions. The first additional security condition is the impossibility of the correct decoding, i.e., the receiver cannot uniquely identify the transmitted message even though the transmitted message is contained in the list. This condition can be trivially satisfied when the transmission rate is larger than the channel capacity. The other additional security condition is the impossibili…
A noisy entropy inequality for boolean functions by Samorodnitsky is applied to binary codes. It is shown that a binary code that achieves capacity on the binary erasure channel admits optimal list size for list decoding on some binary symmetric channels (in a regime where this optimal list size is exponentially large).
An efficient implementation of min-sum SC/list decoding of convolutional polar codes is proposed. The complexity of the proposed implementation of SC decoding is more than two times smaller than the straightforward implementation. Moreover, the proposed list decoding algorithm does not require to copy any LLRs during decoding.
List-decoding and list-recovery are important generalizations of unique decoding that received considerable attention over the years. However, the optimal trade-off among list-decoding (resp. list-recovery) radius, list size, and the code rate are not fully understood in both problems. This paper takes a step towards this direction when the list size is a given constant and the alphabet size is large (as a function of the code length). We prove a new Singleton-type upper bound for list-decodable codes, which improves upon the previously known bound by roughly a factor of $1/L$, where $L$ is th…
We derive the Wu list-decoding algorithm for Generalised Reed-Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm (EA) as the initial algorithm instead of the Berlekamp-Massey algorithm (BMA). We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary Goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami-Sudan algorithm (GSA), immediately leading to equality in the decod…
We present a list decoding algorithm for quaternary negacyclic codes over the Lee metric. To achieve this result, we use a Sudan-Guruswami type list decoding algorithm for Reed-Solomon codes over certain ring alphabets. Our decoding strategy for negacyclic codes over the ring $\mathbb Z_4$ combines the list decoding algorithm by Wu with the Gröbner basis approach for solving a key equation due to Byrne and Fitzpatrick.
A latest coding scheme named polarization-adjusted convolutional (PAC) codes is shown to approach the dispersion bound for the code (128,64) under list decoding. However, to achieve the near-bound performance, the list size of list decoding needs to be excessively large, which leads to insufferable latency. In this paper, to improve the speed of list decoding, fast list decoders for PAC codes are proposed. We define four types of constituent nodes and provide fast list decoding algorithms for each of them. Simulation results present that fast list decoding with three types of constituent nodes…
List decoding of insertions and deletions in the Levenshtein metric is considered. The Levenshtein distance between two sequences is the minimum number of insertions and deletions needed to turn one of the sequences into the other. In this paper, a Johnson-like upper bound on the maximum list size when list decoding in the Levenshtein metric is derived. This bound depends only on the length and minimum Levenshtein distance of the code, the length of the received word, and the alphabet size. It shows that polynomial-time list decoding beyond half the Levenshtein distance is possible for many pa…
This paper is concerned with list decoding of $2$-interleaved binary alternant codes. The principle of the proposed algorithm is based on a combination of a list decoding algorithm for (interleaved) Reed-Solomon codes and an algorithm for (non-interleaved) alternant codes. A new upper bound on the decoding radius is derived and the list size is shown to scale polynomially in the code parameters. While it remains an open problem whether this upper bound is achievable, the provided simulation results show that a decoding radius exceeding the binary Johnson radius can be achieved with a high prob…
Different variants of the code-based McEliece cryptosystem were pro- posed to reduce the size of the public key. All these variants use very structured codes, which open the door to new attacks exploiting the underlying structure. In this paper, we show that the dyadic variant can be designed to resist all known attacks. In light of a new study on list decoding algorithms for binary Goppa codes, we explain how to increase the security level for given public keysizes. Using the state-of-the-art list decoding algorithm instead of unique decoding, we exhibit a keysize gain of about 4% for the sta…
Understanding the limits of list-decoding and list-recovery of Reed-Solomon (RS) codes is of prime interest in coding theory and has attracted a lot of attention in recent decades. However, the best possible parameters for these problems are still unknown, and in this paper, we take a step in this direction. We show the existence of RS codes that are list-decodable or list-recoverable beyond the Johnson radius for \emph{any} rate, with a polynomial field size in the block length. In particular, we show that for any $ε\in (0,1)$ there exist RS codes that are list-decodable from radius $1-ε$ and…
A heuristic construction of polar codes for successive cancellation list (SCL) decoding with a given list size is proposed to balance the trade-off between performance measured in frame error rate (FER) and decoding complexity. Furthermore, a construction based on dynamically frozen bits with constraints among the "low weight bits" (LWB) is presented. Simulation results show that the LWB-polar codes outperform the CRC-polar codes and the eBCH-polar codes under SCL decoding.
An open question about Gabidulin codes is whether polynomial-time list decoding beyond half the minimum distance is possible or not. In this contribution, we give a lower and an upper bound on the list size, i.e., the number of codewords in a ball around the received word. The lower bound shows that if the radius of this ball is greater than the Johnson radius, this list size can be exponential and hence, no polynomial-time list decoding is possible. The upper bound on the list size uses subspace properties.
We extend the notion of list decoding to {\em ratio list decoding} which involves a list decoder whose list size is specified as a function of the number of messages $M_n$ and the block length $n$. We present necessary and sufficient conditions on $M_n$ for the existence of code sequences which enable reliable list decoding with respect to the desired list size $L(M_n,n)$. It is shown that the ratio-capacity, defined as the supremum of achievable normalized logarithms of the ratio $r(M_n,n)=M_n/L(M_n,n)$ is equal to the Shannon channel capacity $C$, for both stochastic and deterministic encodi…
We show that locally repairable codes (LRCs) can be list decoded efficiently beyond the Johnson radius for a large range of parameters by utilizing the local error correction capabilities. The new decoding radius is derived and the asymptotic behavior is analyzed. We give a general list decoding algorithm for LRCs that achieves this radius along with an explicit realization for a class of LRCs based on Reed-Solomon codes (Tamo-Barg LRCs). Further, a probabilistic algorithm for unique decoding of low complexity is given and its success probability analyzed.
We consider a decoder with an erasure option and a variable size list decoder for channels with non-casual side information at the transmitter. First, universally achievable error exponents are offered for decoding with an erasure option using a parameterized decoder in the spirit of Csiszár and Körner's decoder. Then, the proposed decoding rule is generalized by extending the range of its parameters to allow variable size list decoding. This extension gives a unified treatment for erasure/list decoding. Exponential bounds on the probability of list error and the average number of incorrect me…
Compared with classical block codes, efficient list decoding of rank-metric codes seems more difficult. Although the list decodability of random rank-metric codes and limits to list decodability have been completely determined, little work on efficient list decoding rank-metric codes has been done. The only known efficient list decoding of rank-metric codes $\mC$ gives decoding radius up to the Singleton bound $1-R-\Ge$ with positive rate $R$ when $ρ(\mC)$ is extremely small, i.e., $Θ(\Ge^2)$ , where $ρ(\mC)$ denotes the ratio of the number of rows over the number of columns of $\mC$ \cite[STO…
Long polar codes can achieve the capacity of arbitrary binary-input discrete memoryless channels under a low complexity successive cancelation (SC) decoding algorithm. But for polar codes with short and moderate code length, the decoding performance of the SC decoding algorithm is inferior. The cyclic redundancy check (CRC) aided successive cancelation list (SCL) decoding algorithm has better error performance than the SC decoding algorithm for short or moderate polar codes. However, the CRC aided SCL (CA-SCL) decoding algorithm still suffer from long decoding latency. In this paper, a reduced…
The Welch--Berlekamp approach for Reed--Solomon (RS) codes forms a bridge between classical syndrome--based decoding algorithms and interpolation--based list--decoding procedures for list size l=1. It returns the univariate error--locator polynomial and the evaluation polynomial of the RS code as a y-root. In this paper, we show the connection between the Welch--Berlekamp approach for a specific Interleaved Reed--Solomon code scheme and the Guruswami--Sudan principle. It turns out that the decoding of Interleaved RS codes can be formulated as a modified Guruswami--Sudan problem with a specific…
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 b…
The problem of source coding with side information (SCSI) is closely related to channel coding. Therefore, existing literature focuses on using the most successful channel codes namely, LDPC codes, turbo codes, and their variants, to solve this problem assuming classical unique decoding of the underlying channel code. In this paper, in contrast to classical decoding, we have taken the list decoding approach. We show that syndrome source coding using list decoding can achieve the theoretical limit. We argue that, as opposed to channel coding, the correct sequence from the list produced by the l…
The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding approaches based on Key Equations and the interpolation-based list decoding algorithms of Guruswami and Sudan for Reed-Solomon codes. The original interpolation conditions of Guruswami and Sudan for Reed-Solomon codes are reformulated in terms of a se…
We propose a low complexity list successive cancellation (LCLSC) decoding algorithm to reduce complexity of traditional list successive cancellation (LSC) decoding of polar codes while trying to maintain the LSC decoding performance at the same time. By defining two thresholds, namely "likelihood ratio (LR) threshold" and "Bhattacharyya parameter threshold", we classify the reliability of each received information bit and the quality of each bit channel. Based on this classification, we implement successive cancellation (SC) decoding instead of LSC decoding when the information bits from "bad"…
Practical random network coding based schemes for multicast include a header in each packet that records the transformation between the sources and the terminal. The header introduces an overhead that can be significant in certain scenarios. In previous work, parity check matrices of error control codes along with error decoding were used to reduce this overhead. In this work we propose novel packet formats that allow us to use erasure decoding and list decoding. Both schemes have a smaller overhead compared to the error decoding based scheme, when the number of sources combined in a packet is…
We consider recursive decoding for Reed-Muller (RM) codes and their subcodes. Two new recursive techniques are described. We analyze asymptotic properties of these algorithms and show that they substantially outperform other decoding algorithms with nonexponential complexity known for RM codes. Decoding performance is further enhanced by using intermediate code lists and permutation procedures. For moderate lengths up to 512, near-optimum decoding with feasible complexity is obtained.
In this work we study the list-decoding size of Reed-Muller codes. Given a received word and a distance parameter, we are interested in bounding the size of the list of Reed-Muller codewords that are within that distance from the received word. Previous bounds of Gopalan, Klivans and Zuckerman \cite{GKZ08} on the list size of Reed-Muller codes apply only up to the minimum distance of the code. In this work we provide asymptotic bounds for the list-decoding size of Reed-Muller codes that apply for {\em all} distances. Additionally, we study the weight distribution of Reed-Muller codes. Prior re…
We focus on classical and quantum list decoding. The capacity of list decoding was obtained by Nishimura in the case when the number of list does not increase exponentially. However, the capacity of the exponential-list case is open even in the classical case while its converse part was obtained by Nishimura. We derive the channel capacities in the classical and quantum case with an exponentially increasing list. The converse part of the quantum case is obtained by modifying Nagaoka's simple proof for strong converse theorem for channel capacity. The direct part is derived by a quite simple ar…
The order statistics based list decoding techniques for linear binary block codes of small to medium block length are investigated. The construction of the list of the test error patterns is considered. The original order statistics decoding is generalized by assuming segmentation of the most reliable independent positions of the received bits. The segmentation is shown to overcome several drawbacks of the original order statistics decoding. The complexity of the order statistics based decoding is further reduced by assuming a partial ordering of the received bits in order to avoid the complex…
A list decoding scheme for universal polar codes is presented. Our scheme applies to the universal polar codes first introduced by Sasoglu and Wang, and generalized to processes with memory by the authors. These codes are based on the concatenation of different polar transforms: a sequence of "slow" transforms and Arikan's original "fast" transform. List decoding of polar codes has been previously presented in the context of the fast transform. However, the slow transform is markedly different and requires new techniques and data structures. We show that list decoding is possible with space co…
This letter proposes successive cancellation list (SCL) decoding of product codes with Reed--Muller (RM) component codes. SCL decoding relies on a product code description based on the $2\times 2$ Hadamard kernel, which enables interpreting the code as an RM subcode. The focus is on a class of product codes considered in wireless communication systems, based on single parity-check and extended Hamming component codes. For short product codes, it is shown that SCL decoding with a moderate list size performs as well as (and, sometimes, outperforms) belief propagation (BP) decoding. Furthermore, …
We study certain combinatorial aspects of list-decoding, motivated by the exponential gap between the known upper bound (of $O(1/γ)$) and lower bound (of $Ω_p(\log (1/γ))$) for the list-size needed to decode up to radius $p$ with rate $γ$ away from capacity, i.e., $1-\h(p)-γ$ (here $p\in (0,1/2)$ and $γ> 0$). Our main result is the following: We prove that in any binary code $C \subseteq \{0,1\}^n$ of rate $1-\h(p)-γ$, there must exist a set $\mathcal{L} \subset C$ of $Ω_p(1/\sqrtγ)$ codewords such that the average distance of the points in $\mathcal{L}$ from their centroid is at most $pn$. In…
List-decoding of Reed-Solomon (RS) codes beyond the so called Johnson radius has been one of the main open questions since the work of Guruswami and Sudan. It is now known by the work of Rudra and Wootters, using techniques from high dimensional probability, that over large enough alphabets most RS codes are indeed list-decodable beyond this radius. In this paper we take a more combinatorial approach which allows us to determine the precise relation (up to the exact constant) between the decoding radius and the list size. We prove a generalized Singleton bound for a given list size, and conjec…
A transform that enables generator-matrix-based Reed-Solomon (RS) coded data to be recovered under interpolation-based list decoding is presented. The transform matrix needs to be computed only once and the transformation of an element from the output list to the desired RS coded data block incurs $k^{2}$ field multiplications, given a code of dimension $k$.
Polar codes achieve outstanding error correction performance when using successive cancellation list (SCL) decoding with cyclic redundancy check. A larger list size brings better decoding performance and is essential for practical applications such as 5G communication networks. However, the decoding speed of SCL decreases with increased list size. Adaptive SCL (ASCL) decoding can greatly enhance the decoding speed, but the decoding latency for each codeword is different so A-SCL is not a good choice for hardware-based applications. In this paper, a hardware-friendly two-staged adaptive SCL (TA…
A list decoding algorithm for matrix-product codes is provided when $C_1,..., C_s$ are nested linear codes and $A$ is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for matrix-product codes with polynomial units.
We present a hardware architecture and algorithmic improvements for list SC decoding of polar codes. More specifically, we show how to completely avoid copying of the likelihoods, which is algorithmically the most cumbersome part of list SC decoding. The hardware architecture was synthesized for a blocklength of N = 1024 bits and list sizes L = 2, 4 using a UMC 90nm VLSI technology. The resulting decoder can achieve a coded throughput of 181 Mbps at a frequency of 459 MHz.
A decoding algorithm for $q$-ary low-density parity-check codes over the $q$-ary symmetric channel is introduced. The exchanged messages are lists of symbols from $\Fq$. A density evolution analysis for maximum list sizes $1$ and $2$ is developed. Thresholds for selected regular low-density parity-check code ensembles are computed showing gains with respect to a similar algorithm in the literature. Finite-length simulation results confirm the asymptotic analysis.
List decoding for arbitrarily varying channels (AVCs) under state constraints is investigated. It is shown that rates within $ε$ of the randomized coding capacity of AVCs with input-dependent state can be achieved under maximal error with list decoding using lists of size $O(1/ε)$. Under average error an achievable rate region and converse bound are given for lists of size $L$. These bounds are based on two different notions of symmetrizability and do not coincide in general. An example is given that shows that for list size $L$ the capacity may be positive but strictly smaller than the random…
We present a new fast Chase decoding algorithm for binary BCH codes. The new algorithm reduces the complexity in comparison to a recent fast Chase decoding algorithm for Reed--Solomon (RS) codes by the authors (IEEE Trans. IT, 2022), by requiring only a single Koetter iteration per edge of the decoding tree. In comparison to the fast Chase algorithms presented by Kamiya (IEEE Trans. IT, 2001) and Wu (IEEE Trans. IT, 2012) for binary BCH codes, the polynomials updated throughout the algorithm of the current paper typically have a much lower degree. To achieve the complexity reduction, we build …
In this paper we propose an enhanced soft cancellation (SCAN) decoder for polar codes based on decoding stages permutation. The proposed soft cancellation list (SCANL) decoder runs $L$ independent SCAN decoders, each one relying on a different permuted factor graph. The estimated bits are selected among the $L$ candidates through a dedicated metric provided by the decoders. Furthermore, we introduce an early-termination scheme reducing decoding latency without affecting error correction performance. We investigate the error-correction performance of the proposed scheme under various combinatio…
In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions of codes known to achieve list-decoding capacity; multivariate multiplicity codes were the first constructions of high-rate locally correctable codes; and univariate multiplicity codes are also known to achieve list-decoding capacity. However, previous analyses of the er…
In a recent paper [1] an improved analysis concerning the analysis of List Decoding was presented. The event that the correct codeword is excluded from the list is central. For the additive white Gaussian noise (AWGN) channel an important quantity is the in [1] called effective Euclidean distance. This was earlier considered in [2] under the name Vector Euclidean Distance, where also a simple mathematical expression for this quantity was easily derived for any list size. In [1], a geometrical analysis gives this when the list size is 1, 2 or 3.
We decode Reed-Solomon codes using soft information provided at the receiver. The Extended Euclidean Algorithm (EEA) is considered as an initial step to obtain an intermediate result. The final decoding result is obtained by interpolating the output of the EEA at the least reliable positions of the received word. We refer to this decoding method as reduced list-decoding, since not all received positions are used in the interpolation as in other list-decoding methods, such as the Guruswami-Sudan and Wu algorithms. Consequently the complexity of the interpolation step is reduced considerably. Th…
We show how Gabidulin codes can be list decoded by using a parametrization approach. For this we consider a certain module in the ring of linearized polynomials and find a minimal basis for this module using the Euclidean algorithm with respect to composition of polynomials. For a given received word, our decoding algorithm computes a list of all codewords that are closest to the received word with respect to the rank metric.
We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes. We show that for {\em every} code, the ratio of its list decoding radius to its minimum distance stays unchanged under the tensor product operation (rather than squaring, as one might expect). This gives the first efficient list decoders and new combinatorial bounds for some natural codes including multivariate polynomials where the degree in each variable is bounded. We show that for {\em every} code, its list decoding radius remains unchanged under $m$…
Linearized Reed-Solomon (LRS) codes are sum-rank metric codes that fulfill the Singleton bound with equality. In the two extreme cases of the sum-rank metric, they coincide with Reed-Solomon codes (Hamming metric) and Gabidulin codes (rank metric). List decoding in these extreme cases is well-studied, and the two code classes behave very differently in terms of list size, but nothing is known for the general case. In this paper, we derive a lower bound on the list size for LRS codes, which is, for a large class of LRS codes, exponential directly above the Johnson radius. Furthermore, we show t…
This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip ($\textsf{XOR}$) channels, erasure channels, $\textsf{AND}$ ($Z$-) channels, $\textsf{OR}$ channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) $(L-1)$-list decodable codes (where the list size $L$ is a universal constant) exist for such channels. Our criterion asserts that: "For any given general adversarial channel, it is possible to construct positive rate $(L-1)$-list decodable codes if and only if the s…
We show how Gabidulin codes can be list decoded by using an iterative parametrization approach. For a given received word, our decoding algorithm processes its entries one by one, constructing four polynomials at each step. This then yields a parametrization of interpolating solutions for the data so far. From the final result a list of all codewords that are closest to the received word with respect to the rank metric is obtained.
The cyclically equivariant neural decoder was recently proposed in [Chen-Ye, International Conference on Machine Learning, 2021] to decode cyclic codes. In the same paper, a list decoding procedure was also introduced for two widely used classes of cyclic codes -- BCH codes and punctured Reed-Muller (RM) codes. While the list decoding procedure significantly improves the Frame Error Rate (FER) of the cyclically equivariant neural decoder, the Bit Error Rate (BER) of the list decoding procedure is even worse than the unique decoding algorithm when the list size is small. In this paper, we propo…
We consider list versions of sparse approximation problems, where unlike the existing results in sparse approximation that consider situations with unique solutions, we are interested in multiple solutions. We introduce these problems and present the first combinatorial results on the output list size. These generalize and enhance some of the existing results on threshold phenomenon and uncertainty principles in sparse approximations. Our definitions and results are inspired by similar results in list decoding. We also present lower bound examples that bolster our results and show they are of …
Lifted Reed-Solomon codes are a natural affine-invariant family of error-correcting codes which generalize Reed-Muller codes. They were known to have efficient local-testing and local-decoding algorithms (comparable to the known algorithms for Reed-Muller codes), but with significantly better rate. We give efficient algorithms for list-decoding and local list-decoding of lifted codes. Our algorithms are based on a new technical lemma, which says that codewords of lifted codes are low degree polynomials when viewed as univariate polynomials over a big field (even though they may be very high de…
List decoding of Hermitian codes is reformulated to allow an efficient and simple algorithm for the interpolation step. The algorithm is developed using the theory of Groebner bases of modules. The computational complexity of the algorithm seems comparable to previously known algorithms achieving the same task, and the algorithm is better suited for hardware implementation.
