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 considered. In particular, the above inequality is usually assumed to hold for any "practical code" due to a debatable application of McMillan's theorem to sources with memory. We thus propose a clarification of the topic, also providing an extended version of McMillan's theorem to be used for Markovian sources.
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…
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…
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{…
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…
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…
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 …
The theory of multichannel prefix codes aims to generalize the classical theory of prefix codes. Although single- and two-channel prefix codes always have decoding trees, the same cannot be said when there are more than two channels. One question is of theoretical interest: Do there exist optimal tree-decodable codes that are not optimal prefix codes? Existing literature, which focused on generalizing single-channel results, covered little about non-tree-decodable prefix codes since they have no single-channel counterparts. In this work, we study the fundamental reason behind the non-tree-deco…
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…
The use of Space-Time Block Codes (STBCs) increases significantly the optimal detection complexity at the receiver unless the low-complexity decodability property is taken into consideration in the STBC design. In this paper we propose a new low-complexity decodable rate-5/4 full-diversity 4 x 4 STBC. We provide an analytical proof that the proposed code has the Non-Vanishing-Determinant (NVD) property, a property that can be exploited through the use of adaptive modulation which changes the transmission rate according to the wireless channel quality. We compare the proposed code to the best e…
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…
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…
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…
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…
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 …
In this article, the first general constructions of fast-decodable, more specifically (conditionally) $g$-group decodable, space-time block codes for the Nonorthogonal Amplify and Forward (NAF) Multiple-Input Multiple-Output (MIMO) relay channel under the half-duplex constraint are proposed. In this scenario, the source and the intermediate relays used for data amplification are allowed to employ multiple antennas for data transmission and reception. The worst-case decoding complexity of the obtained codes is reduced by up to $75%$. In addition to being fast-decodable, the proposed codes achie…
Space-time coding techniques have become common-place in wireless communication standards as they provide an effective way to mitigate the fading phenomena inherent in wireless channels. However, the use of Space-Time Block Codes (STBCs) increases significantly the optimal detection complexity at the receiver unless the low complexity decodability property is taken into consideration in the STBC design. In this letter we propose a new low-complexity decodable rate-1 full-diversity 4 x 4 STBC. We provide an analytical proof that the proposed code has the Non-Vanishing-Determinant (NVD) property…
Multiple Input Double Output (MIDO) asymmetric space-time codes for 4 transmit antennas and 2 receive antennas can be employed in the downlink from base stations to portable devices. Previous MIDO code constructions with low Maximum Likelihood (ML) decoding complexity, full diversity and the non-vanishing determinant (NVD) property are mostly based on cyclic division algebras. In this paper, a new family of MIDO codes with the NVD property based on crossed-product algebras over Q is introduced. Fast decodability follows naturally from the structure of the codewords which consist of four genera…
A set of linearly constrained permutation matrices are proposed for constructing a class of permutation codes. Making use of linear constraints imposed on the permutation matrices, we can formulate a minimum Euclidian distance decoding problem for the proposed class of permutation codes as a linear programming (LP) problem. The main feature of this class of permutation codes, called LP decodable permutation codes, is this LP decodability. It is demonstrated that the LP decoding performance of the proposed class of permutation codes is characterized by the vertices of the code polytope of the c…
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 …
An index code for broadcast channel with receiver side information is locally decodable if each receiver can decode its demand by observing only a subset of the transmitted codeword symbols instead of the entire codeword. Local decodability in index coding is known to reduce receiver complexity, improve user privacy and decrease decoding error probability in wireless fading channels. Conventional index coding solutions assume that the receivers observe the entire codeword, and as a result, for these codes the number of codeword symbols queried by a user per decoded message symbol, which we ref…
In this paper, we propose a fast decoder algorithm for uniquely decodable (errorless) code sets for overloaded synchronous optical code-division multiple-access (O-CDMA) systems. The proposed decoder is designed in a such a way that the users can uniquely recover the information bits with a very simple decoder, which uses only a few comparisons. Compared to maximum-likelihood (ML) decoder, which has a high computational complexity for even moderate code lengths, the proposed decoder has much lower computational complexity. Simulation results in terms of bit error rate (BER) demonstrate that th…
Non-uniquely decodable codes can be defined as the codes that cannot be uniquely decoded without additional disambiguation information. These are mainly the class of non-prefix-free codes, where a codeword can be a prefix of other(s), and thus, the codeword boundary information is essential for correct decoding. Although the codeword bit stream consumes significantly less space when compared to prefix--free codes, the additional disambiguation information makes it difficult to catch the performance of prefix-free codes in total. Previous studies considered compression with non-prefix-free code…
In this paper, we consider the problem of recursively designing uniquely decodable ternary code sets for highly overloaded synchronous code-division multiple-access (CDMA) systems. The proposed code set achieves larger number of users $K < K_{max}^t$ than any other known state-of-the-art ternary codes that offer low-complexity decoders in the noisy transmission. Moreover, we propose a simple decoder that uses only a few comparisons and can allow the user to uniquely recover the information bits. Compared to maximum likelihood (ML) decoder, which has a high computational complexity for even mod…
In this paper, collocated and distributed space-time block codes (DSTBCs) which admit multi-group maximum likelihood (ML) decoding are studied. First the collocated case is considered and the problem of constructing space-time block codes (STBCs) which optimally tradeoff rate and ML decoding complexity is posed. Recently, sufficient conditions for multi-group ML decodability have been provided in the literature and codes meeting these sufficient conditions were called Clifford Unitary Weight (CUW) STBCs. An algebraic framework based on extended Clifford algebras is proposed to study CUW STBCs …
We investigate the ratio $ρ_{n,L}$ of prefix codes to all uniquely decodable codes over an $n$-letter alphabet and with length distribution $L$. For any integers $n\geq 2$ and $m\geq 1$, we construct a lower bound and an upper bound for $\inf_Lρ_{n,L}$, the infimum taken over all sequences $L$ of length $m$ for which the set of uniquely decodable codes with length distribution $L$ is non-empty. As a result, we obtain that this infimum is always greater than zero. Moreover, for every $m\geq 1$ it tends to 1 when $n\to\infty$, and for every $n\geq 2$ it tends to 0 when $m\to\infty$. In the case …
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…
Two sets $A, B \subseteq \{0, 1\}^n$ form a Uniquely Decodable Code Pair (UDCP) if every pair $a \in A$, $b \in B$ yields a distinct sum $a+b$, where the addition is over $\mathbb{Z}^n$. We show that every UDCP $A, B$, with $|A| = 2^{(1-ε)n}$ and $|B| = 2^{βn}$, satisfies $β\leq 0.4228 +\sqrtε$. For sufficiently small $ε$, this bound significantly improves previous bounds by Urbanke and Li~[Information Theory Workshop '98] and Ordentlich and Shayevitz~[2014, arXiv:1412.8415], which upper bound $β$ by $0.4921$ and $0.4798$, respectively, as $ε$ approaches $0$.
For every natural number $n\geq 2$ and every finite sequence $L$ of natural numbers, we consider the set $UD_n(L)$ of all uniquely decodable codes over an $n$-letter alphabet with the sequence $L$ as the sequence of code word lengths, as well as its subsets $PR_n(L)$ and $FD_n(L)$ consisting of, respectively, the prefix codes and the codes with finite delay. We derive the estimation for the quotient $|UD_n(L)|/|PR_n(L)|$, which allows to characterize those sequences $L$ for which the equality $PR_n(L)=UD_n(L)$ holds. We also characterize those sequences $L$ for which the equality $FD_n(L)=UD_n…
In this paper, we consider the so-called uniquely decodable one-to-one code (UDOOC) that is formed by inserting a "comma" indicator, termed the unique word (UW), between consecutive one-to-one codewords for separation. Along this research direction, we first investigate several general combinatorial properties of UDOOCs, in particular the enumeration of the number of UDOOC codewords for any (finite) codeword length. Based on the obtained formula on the number of length-n codewords for a given UW, the per-letter average codeword length of UDOOC for the optimal compression of a given source stat…
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-…
Space-Time Block Codes (STBCs) suffer from a prohibitively high decoding complexity unless the low-complexity decodability property is taken into consideration in the STBC design. For this purpose, several families of STBCs that involve a reduced decoding complexity have been proposed, notably the multi-group decodable and the fast decodable (FD) codes. Recently, a new family of codes that combines both of these families namely the fast group decodable (FGD) codes was proposed. In this paper, we propose a new construction scheme for rate-1 FGD codes for 2^a transmit antennas. The proposed sche…
In this work, a new fast-decodable space-time block code (STBC) is proposed. The code is full-rate and full-diversity for 4x2 multiple-input multiple-output (MIMO) transmission. Due to the unique structure of the codeword, the proposed code requires a much lower computational complexity to provide maximum-likelihood (ML) decoding performance. It is shown that the ML decoding complexity is only O(M^{4.5}) when M-ary square QAM constellation is used. Finally, the proposed code has highest minimum determinant among the fast-decodable STBCs known in the literature. Simulation results prove that th…
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…
Grant-free multiple-access (GFMA) is a valuable research topic, since it can support multiuser transmission with low latency. This paper constructs novel uniquely-decodable multi-amplitude sequence (UDAS) sets for GFMA systems, which can provide high spectrum efficiency (SE) with low-complexity active user detection (AUD) algorithm. First of all, we propose an UDAS-based multi-dimensional bit interleaving coded modulation (MD-BICM) transmitter; then introduce the definition of UDAS and construct two kinds of UDAS sets based on cyclic and quasi-cyclic matrix modes. Besides, we present a statist…
In this work, we give the first construction of high-rate locally list-recoverable codes. List-recovery has been an extremely useful building block in coding theory, and our motivation is to use these codes as such a building block. In particular, our construction gives the first capacity-achieving locally list-decodable codes (over constant-sized alphabet); the first capacity achieving globally list-decodable codes with nearly linear time list decoding algorithm (once more, over constant-sized alphabet); and a randomized construction of binary codes on the Gilbert-Varshamov bound that can be …
In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are permutations of a multiset that generally consist of duplicate entries. We first introduce a class of binary matrices called multipermutation matrices, each of which corresponds to a unique and distinct multipermutation. By enforcing a set of linear constraints on these matrices, we define a new class of codes that we term LP-decodable multipermutation codes. In order to decode these codes using a linear program (LP), thereby enabling soft decoding, we characterize the convex hull o…
This paper addresses the problem of reducing the delivery time of data messages to cellular users using instantly decodable network coding (IDNC) with physical-layer rate awareness. While most of the existing literature on IDNC does not consider any physical layer complications and abstract the model as equally slotted time for all users, this paper proposes a cross-layer scheme that incorporates the different channel rates of the various users in the decision process of both the transmitted message combinations and the rates with which they are transmitted. The consideration of asymmetric rat…
The notion of universally decodable matrices (UDMs) was recently introduced by Tavildar and Viswanath while studying slow fading channels. It turns out that the problem of constructing UDMs is tightly connected to the problem of constructing maximum distance separable (MDS) codes. In this paper, we first study the properties of UDMs in general and then we discuss an explicit construction of a class of UDMs, a construction which can be seen as an extension of Reed-Solomon codes. In fact, we show that this extension is, in a sense to be made more precise later on, unique. Moreover, the structure…
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…
We consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter $p$ and (b) the process which adds the errors can be described by a sufficiently simple circuit. Codes for such channel models are attractive since, like codes for standard adversarial errors, they can handle channels whose true behavior is unknown or varying over time. For two classes of channels, we provide explicit, efficiently encodable/decodable codes of optimal rate where only inefficiently de…
We construct an explicit family of linear rank-metric codes over any field ${\mathbb F}_h$ that enables efficient list decoding up to a fraction $ρ$ of errors in the rank metric with a rate of $1-ρ-ε$, for any desired $ρ\in (0,1)$ and $ε> 0$. Previously, a Monte Carlo construction of such codes was known, but this is in fact the first explicit construction of positive rate rank-metric codes for list decoding beyond the unique decoding radius. Our codes are subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly …
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…
