Myopic arbitrarily varying channels (AVCs) are point-to-point communication models in which a channel state is controlled by a malicious adversary (a jammer) who receives side-information about the transmitted codeword via a side-channel (wiretapping) and wishes to maximize the probability of error. Compared to standard "oblivious" AVCs, myopic AVCs can potentially use the side information to launch a more effective attack, lowering the capacity of the channel. In this paper, we define a novel property, myopic symmetrizability, and prove it is a sufficient condition for the capacity of any myopic AVC to be zero. We also study the sufficiently myopic setting, in which, roughly speaking, the jammer's side information reveals less information on the codeword transmitted than eventually available at the receiver. In this scenario we show that myopic symmetrizability is also a necessary condition for the capacity to equal zero, by providing a novel code construction using non-i.i.d. codebooks. A key technical lemma, interesting in its own right, is an argument showing that for any positive-rate code (whether for myopic AVCs or not) one can identify a corresponding distribution PX,X'that is a convex combination of product distributions, and such that a constant fraction of pairs of codewords have an empirical distribution approximately equaling PX,X'.

The difference in capacity formulae between worst-case and average-case channel noise models has been part of information theory since the early days of the field. This paper continues a line of work studying intermediate models in which the channel behavior can depend partially on the transmitted codeword. In particular, we consider a model in which a binary erasure channel (with maximum fraction of erasures p) is controlled by an adversary who can observe the transmitted codeword through an independent and memoryless erasure channel (with erasure probability q). Upper and lower bounds on the capacity are given for two models: a noncausal model, in which the adversary can choose their erasures based on the entire (partially observed) codeword, and a causal model, in which at each time the adversary must choose its erasures based on the current and previously observed codeword bits. The achievable rate for the noncausal case is larger than the Gilbert-Varshamov bound and for some parameter ranges exceeds the linear programming (LP) bound; we also provide a non-trivial outer bound on the capacity. For the causal case, we show the capacity is 1-2p+q for p ≥ q (prior work shows the capacity to equal 1-p when p<;q). Our code construction in both scenarios are novel, requiring the encoder to carefully add “low-weight correlated noise” to its transmission.

The fundamental limits of communication over state-dependent discrete memoryless channels with noiseless feedback are studied, under the assumption that the communicating parties are allowed to use variable-length coding schemes. Various cases are analyzed, with the employed coding schemes having either bounded or unbounded codeword lengths, and with state information revealed to the encoder and/or decoder in a strictly causal, causal, or non-causal manner. In each of these settings, necessary and sufficient conditions for positivity of the zero-error capacity are obtained and it is shown that, whenever the zero-error capacity is positive, it equals the conventional vanishing-error capacity. Moreover, it is shown that the vanishing-error capacity of state-dependent channels is not increased by the use of feedback and variable-length coding. Both these kinds of capacities of state-dependent channels with feedback are thus fully characterized.

We establish a general formula for the maximum size of finite length block codes with minimum pairwise distance no less than d. The achievability argument involves an iterative construction of a set of radius-d balls, each centered at a codeword. We demonstrate that the number of such balls that cover the entire code space cannot exceed this maximum size. Our approach can be applied to codes i) with elements over arbitrary code alphabets, and ii) under a broad class of distance measures. Our formula indicates that the maximum code size can be fully characterized by the cumulative distribution function of the distance measure evaluated at two independent and identically distributed random codewords. When the two random codewords assume a uniform distribution over the entire code alphabet, our formula recovers and thus naturally generalizes the Gilbert-Varshamov (GV) lower bound. Finally, we extend our study to the asymptotic setting.

The zero-error capacity of state-dependent channels with noiseless feedback is determined, under the assumption that the transmitter and the receiver are allowed to use variable-length coding schemes. Various cases are analyzed, with the employed coding schemes having either bounded or unbounded codeword lengths and with state information revealed to the encoder and/or decoder in a strictly causal, causal, or noncausal manner. In each of these settings, necessary and sufficient conditions for positivity of the zero-error capacity are obtained and it is shown that, whenever the zero-error capacity is positive, it equals the conventional vanishing-error capacity. A comparison of the results with the recently solved fixed-length case is given.

In this paper, an exact distance spectrum formula for the largest minimum Hamming distance of finite-length binary block codes is presented. The exact formula indicates that the largest minimum distance of finite-length block codes can be fully characterized by the information spectrum of the Hamming distance between two independent and identically distributed (i.i.d.) random codewords. The distance property of finite-length block codes is then connected to the distance spectrum. A side result of this work is a new lower bound to the largest minimum distance of finite-length block codes. Numerical examinations show that the new lower bound improves the finite-length Gilbert-Varshamov lower bound and can reach the minimum distance of existing finite-length block codes.

In this paper, we give a framework for constructing codes over the binary energy harvesting channel when the energy arrivals are random and the battery has large but finite size. We study both noiseless and noisy binary channels (i.e., bit flips). In the noiseless case, we present an encoding strategy (called exponential backoff encoding) which uses a decreasing amount of energy in consecutive transmissions between energy arrivals. We analyze the achievable rate of backoff encoding and show that it can outperform a uniform energy usage policy. We then extend the encoding strategy to a noisy binary channel, suggest a corresponding decoder, and analyze its performance. We believe our constructive approach complements existing approaches which focus on the information capacity of energy harvesting channels.

The explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Yet, the codes being deployed in practice are fairly short. In this work, we address what we view as the main coding theoretic barrier to deploying longer codes in storage: at large lengths, failures are not independent and correlated failures are inevitable. This motivates designing codes that allow quick data recovery even after large correlated failures, and which have efficient encoding and decoding. We propose that code design for distributed storage be viewed as a two step process. The first step is choose a topology of the code, which incorporates knowledge about the correlate d failures that need to be handled, and ensures local recovery from such failures. In the second step one specifies a code with the chosen topology by choosing coefficients from a finite field Fq. In this step, one tries to balance reliability (which is better over larger fields) with encoding and decoding efficiency (which is better over smaller fields). This work initiates an in-depth study of this reliability/efficiency tradeoff. We consider the field-size needed for achieving maximal recover ability: the strongest reliability possible with a given topology. We propose a family of topologies called grid-like topologies which unify a number of topologies considered both in theory and practice, and prove the following results about codes for such topologies: The first super-polynomial lower bound on the field size needed for achieving maximal recoverability in a simple grid-like topology. To our knowledge, there was no super-linear lower bound known before, for any topology. A combinatorial characterization of erasure patterns correctable by Maximally Recoverable codes for a topology which corresponds to tensoring MDS codes with a parity check code. This topology is used in practice (for instance see [MLR+14]). We conjecture a similar characterization for Maximally Recoverable codes instantiating arbitrary tensor product topologies.

The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst case deletions, with a focus on constructing efficiently decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following tradeoffs (for any ε > 0): 1) codes that can correct a fraction 1 - ε of deletions with rate poly(ε) over an alphabet of size poly(1/ε); 2) binary codes of rate 1-Õ(√ε) that can correct a fraction ε of deletions; and 3) Binary codes that can be list-decoded from a fraction (1/2-ε) of deletions with rate poly(ε). This paper gives the first efficient constructions which meet the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate approaching 1 over a fixed alphabet. The abovementioned results bring our understanding of deletion code constructions in these regimes to a similar level as worst case errors.

We construct an explicit family of Fh-linear rankmetric codes over any field Fh that enables efficient list-decoding up to a fraction p of errors in the rank metric with a rate of 1 - ρ - e, for any desired ρ ∈ (0, 1) and e > 0. This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an Fh-subspace that evades the structured subspaces over an extension field Fht that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that listdecoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes.

Locally testable codes (LTCs) of constant minimum (absolute) distance that allow the tester to make a nearly linear number of queries have become the focus of attention recently due to their connections to central questions in approximability theory. In particular, the binary Reed-Muller code of block length N and absolute distance d is known to be testable with O(N/d) queries, and has a dimension of  N – (log N)log d. The polylogarithmically small co-dimension is the basis of constructions of small set expanders with many “bad” eigenvalues, and size-efficient PCPs based on a shorter version of the long code. The smallest possible co-dimension for a distance d code (without any testability requirement) is , achieved by BCH codes. This raises the natural question of understanding where in the spectrum between the two classical families, Reed-Muller and BCH, the optimal co-dimension of a distance d LTC lies — in other words the “price” one has to pay for local testability. One promising approach for constructing LTCs is to focus on affine-invariant codes, whose structure makes testing guarantees easier to deduce than for general codes. Along these lines, the authors of [HRZS13] and [GKS13] recently constructed an affine-invariant family of high-rate LTCs with slightly smaller co-dimension than Reed-Muller codes. In this work, we show that their construction is essentially optimal among linear affine-invariant LTCs that contain the Reed-Muller code of the appropriate degree.

We construct an explicit family of linear rank-metric codes over any field F that enables efficient list decoding up to a fraction rho of errors in the rank metric with a rate of 1-rho-eps, for any desired rho in (0,1) and eps > 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 explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an F-subspace that evades certain structured subspaces over an extension field of F. These structured spaces arise from the linear-algebraic list decoder for Gabidulin codes due to Guruswami and Xing (STOC'13). Our construction is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which are a collection of subspaces, every pair of which have low-dimensional intersection, and which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order that are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list decoding RS codes reduces to list decoding such folded RS codes. However, as we only list decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list decoding RS codes.

Folded Reed-Solomon (RS) codes are an explicit family of codes that achieve the optimal tradeoff between rate and list error-correction capability: specifically, for any ε > 0, Guruswami and Rudra presented annO(1/ ε)time algorithm to list decode appropriate folded RS codes of rateRfrom a fraction 1-R-ε of errors. The algorithm is based on multivariate polynomial interpolation and root-finding over extension fields. It was noted by Vadhan that interpolating a linear polynomial suffices for a statement of the above form. Here, we give a simple linear-algebra-based analysis of this variant that eliminates the need for the computationally expensive root-finding step over extension fields (and indeed any mention of extension fields). The entire list-decoding algorithm is linear-algebraic, solving one linear system for the interpolation step, and another linear system to find a small subspace of candidate solutions. Except for the step of pruning this subspace, the algorithm can be implemented to run in quadratic time. We also consider a closely related family of codes, called (orderm) derivative codes and defined over fields of large characteristic, which consist of the evaluations offas well as its firstm-1 formal derivatives atNdistinct field elements. We show how our linear-algebraic methods for folded RS codes can be used to show that derivative codes can also achieve the above optimal tradeoff. The theoretical drawback of our analysis for folded RS codes and derivative codes is that both the decoding complexity and proven worst-case list-size bound arenΩ(1/ ε). By combining the above idea with a pseudorandom subset of all polynomials as messages, we get a Monte Carlo construction achieving a list-size bound ofO(1/ ε2) which is quite close to the existentialO(1/ ε) bound (however, the decoding complexity remainsnΩ(1/ ε)). Our work highlights that constructing an explicit subspace-evasive subset that has small intersection with low-dimensional subspaces-an interesting problem in pseudorandomness in its own right-could lead to explicit codes with better list-decoding guarantees.

We present a construction of subspace codes along with an efficient algorithm for list decoding from both insertions and deletions, handling an information-theoretically maximum fraction of these with polynomially small rate. Our construction is based on a variant of the folded Reed-Solomon codes in the world of linearized polynomials, and the algorithm is inspired by the recent linear-algebraic approach to list decoding [4]. Ours is the first list decoding algorithm for subspace codes that can handle deletions; even one deletion can totally distort the structure of the basis of a subspace and is thus challenging to handle. When there are only insertions, we also present results for list decoding subspace codes that are the linearized analog of Reed-Solomon codes (proposed in [15, 8], and closely related to the Gabidulin codes for rank-metric coding), obtaining some improvements over similar results in [10].

Conversation between two individuals requires verbal dialogue; the majority of human communication however consists of non-verbal cues such as gestures and facial expressions. Blind individuals are thus hindered in their interaction capabilities. To address this, we are building a computer vision system with facial recognition and expression algorithms to relay nonverbal messages to a blind user. The device will communicate the identities and facial expressions of communication partners in realtime. In order to ensure that this device will be useful to the blind community, we conducted surveys and interviews and we are working with subjects to test prototypes of the device. This paper describes the algorithms and design concepts incorporated in this device, and it provides a commentary on early survey and interview results. A corresponding poster with demonstration stills is exhibited at this conference.