We initiate a study of locally decodable codes with randomized encoding. Standard locally decodable codes are error correcting codes with a deterministic encoding function and a randomized decoding function, such that any desired message bit can be recovered with good probability by querying only a small number of positions in the corrupted codeword. This allows one to recover any message bit very efficiently in sub-linear or even logarithmic time. Besides this straightforward application, locally decodable codes have also found many other applications such as private information retrieval, secure multiparty computation, and average-case complexity.
However, despite extensive research, the tradeoff between the rate of the code and the number of queries is somewhat disappointing. For example, the best known constructions still need super-polynomially long codeword length even with a logarithmic number of queries, and need a polynomial number of queries to achieve a constant rate. In this paper, we show that by using a randomized encoding, in several models we can achieve significantly better rate-query tradeoff. In addition, our codes work for both the standard Hamming errors, and the more general and harder edit errors.
We consider the locality of encoding and decoding operations in distributed storage systems (DSS), and propose a new class of codes, called locally encodable and decodable codes (LEDC), that provides a higher degree of operational locality compared to currently known codes. For a given locality structure, we derive an upper bound on the global distance and demonstrate the existence of an optimal LEDC for sufficiently large field size. In addition, we also construct two families of optimal LEDC for fields with size linear in code length.
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…
In the random deletion channel, each bit is deleted independently with probability $p$. For the random deletion channel, the existence of codes of rate $(1-p)/9$, and thus bounded away from $0$ for any $p < 1$, has been known. We give an explicit construction with polynomial time encoding and deletion correction algorithms with rate $c_0 (1-p)$ for an absolute constant $c_0 > 0$.
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 …
Maximum run-length limited codes are constraint codes used in communication and data storage systems. Insertion/deletion correcting codes correct insertion or deletion errors caused in transmitted sequences and are used for combating synchronization errors. This paper investigates the maximum run-length limited single insertion/deletion correcting (RLL-SIDC) codes. More precisely, we construct efficiently encodable and decodable RLL-SIDC codes. Moreover, we present its encoding algorithm and show the redundancy of the code.
Recent efforts in coding theory have focused on building codes for insertions and deletions, called insdel codes, with optimal trade-offs between their redundancy and their error-correction capabilities, as well as efficient encoding and decoding algorithms. In many applications, polynomial running time may still be prohibitively expensive, which has motivated the study of codes with super-efficient decoding algorithms. These have led to the well-studied notions of Locally Decodable Codes (LDCs) and Locally Correctable Codes (LCCs). Inspired by these notions, Ostrovsky and Paskin-Cherniavsky (…
Index coding achieves bandwidth savings by jointly encoding the messages demanded by all the clients in a broadcast channel. The encoding is performed in such a way that each client can retrieve its demanded message from its side information and the broadcast codeword. In general, in order to decode its demanded message symbol, a receiver may have to observe the entire transmitted codeword. Querying or downloading the codeword symbols might involve costs to a client -- such as network utilization costs and storage requirements for the queried symbols to perform decoding. In traditional index c…
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…
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…
For a family/sequence of STBCs $\mathcal{C}_1,\mathcal{C}_2,\dots$, with increasing number of transmit antennas $N_i$, with rates $R_i$ complex symbols per channel use (cspcu), the asymptotic normalized rate is defined as $\lim_{i \to \infty}{\frac{R_i}{N_i}}$. A family of STBCs is said to be asymptotically-good if the asymptotic normalized rate is non-zero, i.e., when the rate scales as a non-zero fraction of the number of transmit antennas, and the family of STBCs is said to be asymptotically-optimal if the asymptotic normalized rate is 1, which is the maximum possible value. In this paper, …
In this paper, we discuss two-stage encoding algorithms capable of correcting a fraction of asymmetric errors. Suppose that the encoder transmits $n$ binary symbols $(x_1,\ldots,x_n)$ one-by-one over the Z-channel, in which a 1 is received only if a 1 is transmitted. At some designated moment, say $n_1$, the encoder uses noiseless feedback and adjusts further encoding strategy based on the partial output of the channel $(y_1,\ldots,y_{n_1})$. The goal is to transmit error-free as much information as possible under the assumption that the total number of errors inflicted by the Z-channel is lim…
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…
Exponential Lower Bounds for 2-query Relaxed Locally Decodable Codes
University of Illinois at Chicago（芝加哥伊利诺伊大学）；Purdue University（普渡大学）；Peking University（北京大学）；University of Waterloo（滑铁卢大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
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…
In this paper, we propose systematic block Markov superposition transmission of repetition~(BMST-R) codes, which can support a wide range of code rates but maintain essentially the same encoding/decoding hardware structure. The systematic BMST-R codes resemble the classical rate-compatible punctured convolutional~(RCPC) codes, except that they are typically non-decodable by the Viterbi algorithm due to the huge constraint length induced by the block-oriented encoding process. The information sequence is partitioned equally into blocks and transmitted directly, while their replicas are interlea…
Randomized (dithered) quantization is a method capable of achieving white reconstruction error independent of the source. Dithered quantizers have traditionally been considered within their natural setting of uniform quantization. In this paper we extend conventional dithered quantization to nonuniform quantization, via a subterfage: dithering is performed in the companded domain. Closed form necessary conditions for optimality of the compressor and expander mappings are derived for both fixed and variable rate randomized quantization. Numerically, mappings are optimized by iteratively imposin…
We present adaptive on-line schemes for lossy encoding of individual sequences under the conditions of the Wyner-Ziv (WZ) problem. In the first part of this article, a set of fixed-rate scalar source codes with zero delay is presented. We propose a randomized on-line coding scheme, which achieves asymptotically (and with high probability), the performance of the best source code in the set, uniformly over all source sequences. The scheme uses the same rate and has zero delay. We then present an efficient algorithm for implementing our on-line coding scheme in the case of a relatively small set…
Polar coding, introduced 2008 by Arikan, is the first (very) efficiently encodable and decodable coding scheme whose information transmission rate provably achieves the Shannon bound for classical discrete memoryless channels in the asymptotic limit of large block sizes. Here we study the use of polar codes for the transmission of quantum information. Focusing on the case of qubit Pauli channels and qubit erasure channels, we use classical polar codes to construct a coding scheme which, using some pre-shared entanglement, asymptotically achieves a net transmission rate equal to the coherent in…
Transmission of information reliably and efficiently across channels is one of the fundamental goals of coding and information theory. In this respect, efficiently decodable deterministic coding schemes which achieve capacity provably have been elusive until as recent as 2008, even though schemes which come close to it in practice existed. This survey tries to give the interested reader an overview of the area. Erdal Arikan came up with his landmark polar coding shemes which achieve capacity on symmetric channels subject to the constraint that the input codewords are equiprobable. His idea is …
We construct explicit algebraic geometry codes built from the Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for alphabet sizes at least 192. Messages are identied with functions in certain Riemann-Roch spaces associated with divisors supported on multiple places. Encoding amounts to evaluating these functions at degree one places. By exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we devise an intricate deterministic ω/2 < 1.19 runtime exponent encoding and 1+ω/2 < 2.19 expected runtime exponent randomized (unique and list) decoding alg…
In this paper, we propose a simple procedure to construct (decodable) good codes with any given alphabet (of moderate size) for any given (rational) code rate to achieve any given target error performance (of interest) over additive white Gaussian noise (AWGN) channels. We start with constructing codes over groups for any given code rates. This can be done in an extremely simple way if we ignore the error performance requirement for the time being. Actually, this can be satisfied by repetition (R) codes and uncoded (UN) transmission along with time-sharing technique. The resulting codes are si…
In this paper, we propose a novel iterative encoding algorithm for DNA storage to satisfy both the GC balance and run-length constraints using a greedy algorithm. DNA strands with run-length more than three and the GC balance ratio far from 50\% are known to be prone to errors. The proposed encoding algorithm stores data at high information density with high flexibility of run-length at most $m$ and GC balance between $0.5\pmα$ for arbitrary $m$ and $α$. More importantly, we propose a novel mapping method to reduce the average bit error compared to the randomly generated mapping method, using …
