We investigate the list decodability of symbol-pair codes in the present paper. Firstly, we show that list decodability of every symbol-pair code does not exceed the Gilbert-Varshamov bound. On the other hand, we are able to prove that with high probability, a random symbol-pair code can be list decoded up to the Gilbert-Varshamov bound. Our second result of this paper is to derive the Johnson-type bound, i.e., a lower bound on list decoding radius in terms of minimum distance. Finally, we present a list decoding algorithm of Reed-Solomon codes beyond the Johnson-type bound.
In 2011, Guruswami-Håstad-Kopparty \cite{Gru} showed that the list-decodability of random linear codes is as good as that of general random codes. In the present paper, we further strengthen the result by showing that the list-decodability of random {\it Euclidean self-orthogonal} codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. Specifically, we show that, for any fixed finite field $\F_q$, error fraction $δ\in (0,1-1/q)$ satisfying $1-H_q(δ)\le \frac12$ and small $ε>0$, with high probability a random Euclidean self-orthogonal code…
In this paper, we analyze the performance of space-time block codes which enable symbolwise maximum likelihood decoding. We derive an upper bound of maximum mutual information (MMI) on space-time block codes that enable symbolwise maximum likelihood decoding for a frequency non-selective quasi-static fading channel. MMI is an upper bound on how much one can send information with vanishing error probability by using the target code.
V. Guruswami and N. Resch prove that the list decodability of $\mathbb{F}_q$-linear rank metric codes is as good as that of random rank metric codes in~\cite{venkat2017}. Due to the potential applications of self-orthogonal rank metric codes, we focus on list decoding of them. In this paper, we prove that with high probability, an $\F_q$-linear self-orthogonal rank metric code over $\mathbb{F}_q^{n\times m}$ of rate $R=(1-τ)(1-\frac{n}{m}τ)-ε$ is shown to be list decodable up to fractional radius $τ\in(0,1)$ and small $ε\in(0,1)$ with list size depending on $τ$ and $q$ at most $O_{τ, q}(\frac{…
It is well known that a random q-ary code of rate Ω(ε^2) is list decodable up to radius (1 - 1/q - ε) with list sizes on the order of 1/ε^2, with probability 1 - o(1). However, until recently, a similar statement about random linear codes has until remained elusive. In a recent paper, Cheraghchi, Guruswami, and Velingker show a connection between list decodability of random linear codes and the Restricted Isometry Property from compressed sensing, and use this connection to prove that a random linear code of rate Ω(ε^2 / log^3(1/ε)) achieves the list decoding properties above, with constant pr…
We introduce a new family of erasure codes, called group decodable code (GDC), for distributed storage system. Given a set of design parameters {α; β; k; t}, where k is the number of information symbols, each codeword of an (α; β; k; t)-group decodable code is a t-tuple of strings, called buckets, such that each bucket is a string of βsymbols that is a codeword of a [β; α] MDS code (which is encoded from αinformation symbols). Such codes have the following two properties: (P1) Locally Repairable: Each code symbol has locality (α; β-α+ 1). (P2) Group decodable: From each bucket we can decode αi…
In this work, we study the problem of list decoding of insertions and deletions. We present a Johnson-type upper bound on the maximum list size. The bound is meaningful only when insertions occur. Our bound implies that there are binary codes of rate $Ω(1)$ that are list-decodable from a $0.707$-fraction of insertions. For any $τ_\mathsf{I} \geq 0$ and $τ_\mathsf{D} \in [0,1)$, there exist $q$-ary codes of rate $Ω(1)$ that are list-decodable from a $τ_\mathsf{I}$-fraction of insertions and $τ_\mathsf{D}$-fraction of deletions, where $q$ depends only on $τ_\mathsf{I}$ and $τ_\mathsf{D}$. We als…
List-decodability of Reed-Solomon codes has received a lot of attention, but the best-possible dependence between the parameters is still not well-understood. In this work, we focus on the case where the list-decoding radius is of the form $r=1-\varepsilon$ for $\varepsilon$ tending to zero. Our main result states that there exist Reed-Solomon codes with rate $Ω(\varepsilon)$ which are $(1-\varepsilon, O(1/\varepsilon))$-list-decodable, meaning that any Hamming ball of radius $1-\varepsilon$ contains at most $O(1/\varepsilon)$ codewords. This trade-off between rate and list-decoding radius is …
$A'(n,d,e)$, the smallest $\ell$ for which every binary error-correcting code of length $n$ and minimum distance $d$ is decodable with a list of size $\ell$ up to radius $e$, is determined for all $d\geq 2e-3$. As a result, $A'(n,d,e)$ is determined for all $e\leq 4$, except for 42 values of $n$.
For every fixed finite field $\F_q$, $p \in (0,1-1/q)$ and $ε> 0$, we prove that with high probability a random subspace $C$ of $\F_q^n$ of dimension $(1-H_q(p)-ε)n$ has the property that every Hamming ball of radius $pn$ has at most $O(1/ε)$ codewords. This answers a basic open question concerning the list-decodability of linear codes, showing that a list size of $O(1/ε)$ suffices to have rate within $ε$ of the "capacity" $1-H_q(p)$. Our result matches up to constant factors the list-size achieved by general random codes, and gives an exponential improvement over the best previously known lis…
In the present paper, we consider list decoding for both random rank metric codes and random linear rank metric codes. Firstly, we show that, for arbitrary $0<R<1$ and $ε>0$ ($ε$ and $R$ are independent), if $0<\frac{n}{m}\leq ε$, then with high probability a random rank metric code in $F_{q}^{m\times n}$ of rate $R$ can be list-decoded up to a fraction $(1-R-ε)$ of rank errors with constant list size $L$ satisfying $L\leq O(1/ε)$. Moreover, if $\frac{n}{m}\geqΘ_R(ε)$, any rank metric code in $F_{q}^{m\times n}$ with rate $R$ and decoding radius $ρ=1-R-ε$ can not be list decoded in ${\rm poly}…
There has been a great deal of work establishing that random linear codes are as list-decodable as uniformly random codes, in the sense that a random linear binary code of rate $1 - H(p) - ε$ is $(p,O(1/ε))$-list-decodable with high probability. In this work, we show that such codes are $(p, H(p)/ε+ 2)$-list-decodable with high probability, for any $p \in (0, 1/2)$ and $ε> 0$. In addition to improving the constant in known list-size bounds, our argument, which is quite simple, works simultaneously for all values of $p$, while previous works obtaining $L = O(1/ε)$ patched together different arg…
Wachter-Zeh in [42], and later together with Raviv [31], proved that Gabidulin codes cannot be efficiently list decoded for any radius $τ$, providing that $τ$ is large enough. Also, they proved that there are infinitely many choices of the parameters for which Gabidulin codes cannot be efficiently list decoded at all. Subsequently, in [41] these results have been extended to the family of generalized Gabidulin codes and to further family of MRD-codes. In this paper, we provide bounds on the list size of rank-metric codes containing generalized Gabidulin codes in order to determine whether or n…
In this paper we propose a revisitation of the topic of unique decodability and of some fundamental theorems of lossless coding. It is widely believed that, for any discrete source X, every "uniquely decodable" block code satisfies E[l(X_1 X_2 ... X_n)]>= H(X_1,X_2,...,X_n), where X_1, X_2,...,X_n are the first n symbols of the source, E[l(X_1 X_2 ... X_n)] is the expected length of the code for those symbols and H(X_1,X_2,...,X_n) is their joint entropy. We show that, for certain sources with memory, the above inequality only holds when a limiting definition of "uniquely decodable code" is co…
In this work, we prove new results concerning the combinatorial properties of random linear codes. Firstly, we prove a lower bound on the list-size required for random linear codes over $\mathbb F_q$ $\varepsilon$-close to capacity to list-recover with error radius $ρ$ and input lists of size $\ell$. We show that the list-size $L$ must be at least $\frac{\log_q\binom{q}{\ell}-R}{\varepsilon}$, where $R$ is the rate of the random linear code. As a comparison, we also pin down the list size of random codes which is $\frac{\log_q\binom{q}{\ell}}{\varepsilon}$. This leaves open the possibility (th…
We analyze the list-decodability, and related notions, of random linear codes. This has been studied extensively before: there are many different parameter regimes and many different variants. Previous works have used complementary styles of arguments---which each work in their own parameter regimes but not in others---and moreover have left some gaps in our understanding of the list-decodability of random linear codes. In particular, none of these arguments work well for list-recovery, a generalization of list-decoding that has been useful in a variety of settings. In this work, we present a …
We show that any q-ary code with sufficiently good distance can be randomly punctured to obtain, with high probability, a code that is list decodable up to radius $1 - 1/q - ε$ with near-optimal rate and list sizes. Our results imply that "most" Reed-Solomon codes are list decodable beyond the Johnson bound, settling the long-standing open question of whether any Reed Solomon codes meet this criterion. More precisely, we show that a Reed-Solomon code with random evaluation points is, with high probability, list decodable up to radius $1 - ε$ with list sizes $O(1/ε)$ and rate $Ω(ε)$. As a secon…
The MDS property (aka the $k$-out-of-$n$ property) requires that if a file is split into several symbols and subsequently encoded into $n$ coded symbols, each being stored in one storage node of a distributed storage system (DSS), then an user can recover the file by accessing any $k$ nodes. We study the so-called $p$-decodable $μ$-secure erasure coding scheme $(1 \leq p \leq k - μ, 0 \leq μ< k, p | (k-μ))$, which satisfies the MDS property and the following additional properties: (P1) strongly secure up to a threshold: an adversary which eavesdrops at most $μ$ storage nodes gains no informati…
We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) a certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of balls in Hamming, Johnson spaces and symmetric groups decay exponentially as their centers drift apart. To verify condition (iii), we prove some large deviation inequalities `on a slice' for functions with Lipschitz conditions. We then use these estimates on intersecti…
Let $k,n,m \in \mathbb{Z}^+$ integers such that $k\leq n \leq m$, let $\mathrm{G}_{n,k}\in \mathbb{F}_{q^m}^n$ be a Delsarte-Gabidulin code. Wachter-Zeh proven that codes belonging to this family cannot be efficiently list decoded for any radius $τ$, providing $τ$ is large enough. This achievement essentially relies on proving a lower bound for the list size of some specific words in $\mathbb{F}_{q^m}^n \setminus \mathrm{G}_{n,k}$. In 2016, Raviv and Wachter-Zeh improved this bound in a special case, i.e. when $n\mid m$. As a consequence, they were able to detect infinite families of Delsarte-…
The list-decodable code has been an active topic in theoretical computer science.There are general results about the list-decodability to the Johnson radius and the list-decoding capacity theorem. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new general simple but strong upper bounds for list-decodable codes in general finite metric spaces based on various covering codes. The general covering code upper bounds can be applied to the case that the volumes of the balls depend on the centers, not only …
We prove that a random linear code over F_q, with probability arbitrarily close to 1, is list decodable at radius (1-1/q-ε) with list size L=O(1/ε^2) and rate R=Ω_q(ε^2/(log^3(1/ε))). Up to the polylogarithmic factor in (1/ε) and constant factors depending on q, this matches the lower bound L=Ω_q(1/ε^2) for the list size and upper bound R=O_q(ε^2) for the rate. Previously only existence (and not abundance) of such codes was known for the special case q=2 (Guruswami, Håstad, Sudan and Zuckerman, 2002). In order to obtain our result, we employ a relaxed version of the well known Johnson bound on…
We study fast lattice decodability of space-time block codes for $n$ transmit and receive antennas, written very generally as a linear combination $\sum_{i=1}^{2l} s_i A_i$, where the $s_i$ are real information symbols and the $A_i$ are $n\times n$ $\mathbb R$-linearly independent complex valued matrices. We show that the mutual orthogonality condition $A_iA_j^* + A_jA_i^*=0$ for distinct basis matrices is not only sufficient but also necessary for fast decodability. We build on this to show that for full-rate ($l = n^2$) transmission, the decoding complexity can be no better than $|S|^{n^2+1}…
In this work, we introduce a framework to study the effect of random operations on the combinatorial list-decodability of a code. The operations we consider correspond to row and column operations on the matrix obtained from the code by stacking the codewords together as columns. This captures many natural transformations on codes, such as puncturing, folding, and taking subcodes; we show that many such operations can improve the list-decoding properties of a code. There are two main points to this. First, our goal is to advance our (combinatorial) understanding of list-decodability, by unders…
Insdel errors occur in communication systems caused by the loss of positional information of the message. Since the work by Guruswami and Wang, there have been some further investigations on the list decoding of insertion codes, deletion codes and insdel codes. However, unlike classical Hamming metric or even rank-metric, there are still many unsolved problems on list decoding of insdel codes. The contributions of this paper mainly consist of two parts. Firstly, we analyze the list decodability of random insdel codes. We show that list decoding of random insdel codes surpasses the Singleton bo…
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…
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…
Polar codes is promising because they can provably achieve the channel capacity while having an explicit construction method. Lots of work have been done for the bit-based decoding algorithm for polar codes. In this paper, generalized symbol-based successive cancellation (SC) and SC list decoding algorithms are discussed. A symbol-based recursive channel combination relationship is proposed to calculate the symbol-based channel transition probability. This proposed method needs less additions than the maximum-likelihood decoder used by the existing symbol-based polar decoding algorithm. In add…
High data-rate Distributed Orthogonal Space-Time Block Codes (DOSTBCs) which achieve the single-symbol decodability and full diversity order are proposed in this paper. An upper bound of the data-rate of the DOSTBC is derived and it is approximately twice larger than that of the conventional repetition-based cooperative strategy. In order to facilitate the systematic constructions of the DOSTBCs achieving the upper bound of the data-rate, some special DOSTBCs, which have diagonal noise covariance matrices at the destination terminal, are investigated. These codes are referred to as the row-mon…
Given a classical error-correcting block code, the task of quantum list decoding is to produce from any quantumly corrupted codeword a short list containing all messages whose codewords exhibit high "presence" in the quantumly corrupted codeword. Efficient quantum list decoders have been used to prove a quantum hardcore property of classical codes. However, the code rates of all known families of efficiently quantum list-decodable codes are, unfortunately, too small for other practical applications. To improve those known code rates, we prove that a specific code family of polynomially small c…
Polar codes are of great interests because they provably achieve the capacity of both discrete and continuous memoryless channels while having an explicit construction. Most existing decoding algorithms of polar codes are based on bit-wise hard or soft decisions. In this paper, we propose symbol-decision successive cancellation (SC) and successive cancellation list (SCL) decoders for polar codes, which use symbol-wise hard or soft decisions for higher throughput or better error performance. First, we propose to use a recursive channel combination to calculate symbol-wise channel transition pro…
An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters [n,k,d] and the list size L. L-MDS codes are then defined as codes that attain this bound (under a slightly stronger notion of list decodability), with 1-MDS codes corresponding to ordinary linear MDS codes. Several properties of such codes are presented; in particular, it is shown that the 2-MDS property is preserved under duality. Finally, explicit constructions for 2-MDS codes are presented through generalized Reed-Solomon (GRS) codes.
Distributed Orthogonal Space-Time Block Codes (DOSTBCs) achieving full diversity order and single-symbol ML decodability have been introduced recently for cooperative networks and an upper-bound on the maximal rate of such codes along with code constructions has been presented. In this report, we introduce a new class of Distributed STBCs called Semi-orthogonal Precoded Distributed Single-Symbol Decodable STBCs (S-PDSSDC) wherein, the source performs co-ordinate interleaving of information symbols appropriately before transmitting it to all the relays. It is shown that DOSTBCs are a special ca…
Distributed Orthogonal Space-Time Block Codes (DOSTBCs) achieving full diversity order and single-symbol ML decodability have been introduced recently by Yi and Kim for cooperative networks and an upperbound on the maximal rate of such codes along with code constructions has been presented. In this paper, we introduce a new class of Distributed STBCs called Semi-orthogonal Precoded Distributed Single-Symbol Decodable STBCs (S-PDSSDC) wherein, the source performs co-ordinate interleaving of information symbols appropriately before transmitting it to all the relays. It is shown that DOSTBCs are …
Linear space-time block codes (STBCs) of unitary rate and full diversity, systematically constructed over arbitrary constellations for any number of transmit antennas are introduced. The codes are obtained by generalizing the existing ABBA STBCs, a.k.a quasi-orthogonal STBCs (QO-STBCs). Furthermore, a fully orthogonal (symbol-by-symbol) decoder for the new generalized ABBA (GABBA) codes is provided. This remarkably low-complexity decoder relies on partition orthogonality properties of the code structure to decompose the received signal vector into lower-dimension tuples, each dependent only on…
We study the list decodability of different ensembles of codes over the real alphabet under the assumption of an omniscient adversary. It is a well-known result that when the source and the adversary have power constraints $ P $ and $ N $ respectively, the list decoding capacity is equal to $ \frac{1}{2}\log\frac{P}{N} $. Random spherical codes achieve constant list sizes, and the goal of the present paper is to obtain a better understanding of the smallest achievable list size as a function of the gap to capacity. We show a reduction from arbitrary codes to spherical codes, and derive a lower…
The classical family of Reed-Solomon codes consist of evaluations of polynomials over the finite field $\mathbb{F}_q$ of degree less than $k$, at $n$ distinct field elements. These are arguably the most widely used and studied codes, as they have both erasure and error-correction capabilities, among many others nice properties. In this survey we study closely related codes, folded Reed-Solomon codes, which are the first constructive codes to achieve the list decoding capacity. We then study two more codes which also have this feature, \textit{multiplicity codes} and \textit{derivative codes}. …
A binary code is said to be a disjunctive list-decoding $s_L$-code (LD $s_L$-code), $s \ge 2$, $L \ge 1$, if the code is identified by the incidence matrix of a family of finite sets in which the union (or disjunctive sum) of any $s$ sets can cover not more than $L-1$ other sets of the family. In this paper, we consider a similar class of binary codes which are based on a {\em symmetric disjunctive sum} (SDS) of binary symbols. By definition, the symmetric disjunctive sum (SDS) takes values from the ternary alphabet $\{0, 1, *\}$, where the symbol~$*$ denotes "erasure". Namely: SDS is equal to…
Existing literature on source coding with side information (SCSI) mostly uses the state-of-the-art channel codes namely LDPC codes, turbo codes, and their variants and assume classical unique decoding. In this paper, we present an algebraic approach to SCSI based on the list decoding of the underlying channel codes. We show that the theoretical limit of SCSI can be achieved in the proposed list decoding based framework when the correlation between the source and side information is $q$-ary symmetric. We argue that, as opposed to channel coding, the correct sequence from the list produced by th…
The Reed-Muller (RM) code encoding $n$-variate degree-$d$ polynomials over ${\mathbb F}_q$ for $d < q$, with its evaluation on ${\mathbb F}_q^n$, has relative distance $1-d/q$ and can be list decoded from a $1-O(\sqrt{d/q})$ fraction of errors. In this work, for $d \ll q$, we give a length-efficient puncturing of such codes which (almost) retains the distance and list decodability properties of the Reed-Muller code, but has much better rate. Specificially, when $q =Ω( d^2/ε^2)$, we given an explicit rate $Ω\left(\fracε{d!}\right)$ puncturing of Reed-Muller codes which have relative distance at…
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.
In this work we consider the list-decodability and list-recoverability of arbitrary $q$-ary codes, for all integer values of $q\geq 2$. A code is called $(p,L)_q$-list-decodable if every radius $pn$ Hamming ball contains less than $L$ codewords; $(p,\ell,L)_q$-list-recoverability is a generalization where we place radius $pn$ Hamming balls on every point of a combinatorial rectangle with side length $\ell$ and again stipulate that there be less than $L$ codewords. Our main contribution is to precisely calculate the maximum value of $p$ for which there exist infinite families of positive rate $…
We present a new information-theoretic definition and associated results, based on list decoding in a source coding setting. We begin by presenting list-source codes, which naturally map a key length (entropy) to list size. We then show that such codes can be analyzed in the context of a novel information-theoretic metric, ε-symbol secrecy, that encompasses both the one-time pad and traditional rate-based asymptotic metrics, but, like most cryptographic constructs, can be applied in non-asymptotic settings. We derive fundamental bounds for ε-symbol secrecy and demonstrate how these bounds can …
We consider the problem of determining the zero-error list-decoding capacity of the $q/(q-1)$ channel studied by Elias (1988). The $q/(q-1)$ channel has input and output alphabet consisting of $q$ symbols, say, $Q = \{x_1,x_2,\ldots, x_q\}$; when the channel receives an input $x \in Q$, it outputs a symbol other than $x$ itself. Let $n(m,q,\ell)$ be the smallest $n$ for which there is a code $C \subseteq Q^n$ of $m$ elements such that for every list $w_1, w_2, \ldots, w_{\ell+1}$ of distinct code-words from $C$, there is a coordinate $j \in [n]$ that satisfies $\{w_1[j], w_2[j], \ldots, w_{\el…
Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed when these recalculations reach the trivial RM codes. In turn, the updated lists of most plausible codewords are used in subsequent decodings. The algorithm is further improved by using permutation techniques on code positions and by eliminating the most error-prone information bits. Simulation results show that for all RM codes of length 256 and many subcodes …
