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A novel scheme is presented for encoding and decoding of Reed-Solomon codes of prime lengths. Encoding is performed on a collection of codewords which are mapped through Galois Fourier transform into a codeword in a low-density parity-check code with a binary parity-check matrix for transmission. Using this matrix, a binary iterative soft-decision decoding algorithm is applied to jointly decode a...
We introduce Cyclone codes which are rateless erasure resilient codes. They combine Pair codes with Luby Transform (LT) codes by computing a code symbol from a random set of data symbols using bitwise XOR and cyclic shift operations. The number of data symbols is chosen according to the Robust Soliton distribution. XOR and cyclic shift operations establish a unitary commutative ring if data symbols...
We prove that the solvability of systems of linear equations and related linear algebraic properties are definable in a fragment of fixed-point logic with counting that only allows polylogarithmically many iterations of the fixed-point operators. This enables us to separate the descriptive complexity of solving linear equations from full fixed-point logic with counting by logical means. As an application...
Recently, Parizi, Telatar, and Merhav [1] determined the exact random coding secrecy exponents for a wiretap channel. In this study, we focus on the computation of such secrecy exponents. To obtain the exact random coding secrecy exponent for a constant composition random coding ensemble, optimization with respect to two stochastic matrices must be performed. Parizi et al. suggested that inner optimization...
Write-once memory (WOM) is a storage device consisting of binary cells which can only increase their levels. A t-write WOM code is a coding scheme which allows to write t times to the WOM without decreasing the levels of the cells. The sum-rate of a WOM code is the ratio between the total number of bits written to the memory and the number of cells. It is known that the maximum sum-rate of a t-write...
Binary AIFV codes, which can attain better compression rate than Huffman codes, uses two code trees that may have incomplete internal nodes, and source symbols are assigned to some internal nodes in addition to leaves. Although the code trees of Huffman codes, which are full binary trees, are well studied, the AIFV code trees have not been yet studied in detail. In this paper, we show that there exists...
The extremely strict code length constraint is the main drawback of lowest density, maximum-distance separable (MDS) array codes of distance greater than 3. To break away from the status quo, we proposed in [5] a family of lowest density MDS array codes of (column) distance 4, called XI-Code. Compared with the previous alternatives, XI-Code has lower encoding and decoding complexities, and much looser...
The capacity region of 1-to-K broadcast packet erasure channels with ACK/NACK is known for some scenarios, e.g., K ≤ 3, etc. However, existing achievability schemes either require knowing the target rate R in advance, and/or have a complicated description of the achievable rate region that is difficult prove whether it matches the capacity or not. This work proposes a new network coding protocol with...
The regular separability problem asks, for two given languages, if there exists a regular language including one of them but disjoint from the other. Our main result is decidability, and PSPACE-completeness, of the regular separability problem for languages of one counter automata without zero tests (also known as one counter nets). This contrasts with undecidability of the regularity problem for...
We consider the problem of constructing codes that can correct δ deletions occurring in an arbitrary binary string of length n bits. Varshamov-Tenengolts (VT) codes can correct all possible single deletions (δ = 1) with an asymptotically optimal redundancy. Finding similar codes for δ ≥ 2 deletions is an open problem. We propose a new family of codes, that we call Guess & Check (GC) codes, that...
Compression of multispectal images is of great importance in an environment where resources such as computational power and memory are scarce. To that end, we propose a new extremely low-complexity encoding approach for compression of multispectral images, that shifts the complexity to the decoding. Our method combines principles from compressed sensing and distributed source coding. Specifically,...
We give a new proof of completeness for the linear-time μ-calculus w.r.t. Kozens's axiomatization. Our proof has the advantage of being constructive, i.e., it builds a proof for every valid formula.
In this paper, we focus on the finite-length performance of spinal codes with a sliding window decoder over binary erasure channel. An expression of the error probability of spinal codes is derived. Particularly, we also derive an expression of the error probability of spinal codes with some known tail bits, which can improve the error-control performance. Moreover, easier-to-compute upper and lower...
Mutually Uncorrelated (MU) codes are a class of codes in which no proper prefix of one codeword is a suffix of another codeword. These codes were originally studied for synchronization purposes and recently, Yazdi et al. showed their applicability to enable random access in DNA storage. In this work we follow the research of Yazdi et al. and study MU codes along with their extensions to correct errors...
This paper is concerned with block Markov super-position transmission of BCH (BMST-BCH) codes. Compared with other BMST codes, BMST-BCH codes can achieve a lower error floor with an encoding memory of two, which is critical to reduce both delay and implementation complexity. To further reduce the implementation complexity, we propose a hard-decision sliding-window decoding algorithm, in which only...
The complexity of software implementations of MDS erasure codes mainly depends on the efficiency of the finite field operations implementation. In this paper, we propose a method to reduce the complexity of the finite field multiplication by using simple transforms between a field and a ring to perform the multiplication in a ring. We show that moving to a ring reduces the complexity of the operations...
We study secure RAID, i.e., low-complexity schemes to store information in a distributed manner that is resilient to node failures and resistant to node eavesdropping. We describe a technique to shorten the secure EVENODD scheme in [6], which can optimally tolerate 2 node failures and 2 eavesdropping nodes. The shortening technique allows us to obtain secure EVENODD schemes of arbitrary lengths, which...
Motivated by the question of whether the recently introduced Reduced Cutset Coding (RCC) [1], [2] offers rate-complexity performance benefits over conventional context-based conditional lossless coding for sources with two-dimensional Markov structure, this paper compares several row-centric coding strategies that vary in the amount of conditioning as well as whether a model or an empirical table...
It was first observed by John Bell that quantum theory predicts correlations between measurement outcomes that lie beyond the explanatory power of local hidden variable theories. These correlations have traditionally been studied extensively in the probabilistic framework. A drawback of this perspective is that one is then forced to use in a single argument the outcomes of mutually-exclusive measurements...
Sparse superposition (SS) codes were originally proposed as a capacity-achieving communication scheme over the additive white Gaussian noise channel (AWGNC) [1]. Very recently, it was discovered that these codes are universal, in the sense that they achieve capacity over any memoryless channel under generalized approximate message-passing (GAMP) decoding [2], although this decoder has never been stated...
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