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Limited magnitude asymmetric error model is well suited for flash memory. In this paper, we consider the construction of asymmetric codes correcting single error over $\mathbb {Z}_{2^{k}r}$ which is based on so called $B_{1}[{4}](2^{k}r)$ set. In fact, we reduce the construction of a maximal size $B_{1}[{4}](2^{k}r)$ set for $k\geq 3$ to the construction of a maximal size $B_{1}[{4}](2^{k-3}r)$ set. Finally, we give an explicit formula of a maximal size $B_{1}[{4}](4r)$ set and some lower bounds of a maximal size $B_{1}[{4}](2r)$ set. By computer searching up to $q\leq 106$ , we conjecture that those lower bounds are tight. [ABSTRACT FROM AUTHOR]

To equip DNA-based data storage with random-access capabilities, Yazdi et al. (2018) prepended DNA strands with specially chosen address sequences called primers and provided certain design criteria for these primers. We provide explicit constructions of error-correcting codes that are suitable as primer addresses and equip these constructions with efficient encoding algorithms. Specifically, our constructions take cyclic or linear codes as inputs and produce sets of primers with similar error-correcting capabilities. Using certain classes of BCH codes, we obtain infinite families of primer sets of length $n$ , minimum distance $d$ with $(d+1) \log _{4}\,\,n +O(1)$ redundant symbols. Our techniques involve reversible cyclic codes (1964), an encoding method of Tavares et al. (1971) and Knuth’s balancing technique (1986). In our investigation, we also construct efficient and explicit binary balanced error-correcting codes and codes for DNA computing. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting codes (abbreviate to EAQECCs) expand the usual paradigm of quantum error correction by allowing two parties to make use of pre-shared entanglement. It is well-known that we can construct an EAQECC from arbitrary classical linear code. In this paper, we construct several classes of entanglement-assisted quantum MDS (EAQMDS) codes by utilizing generalized Reed-Solomon (GRS) codes. The main contribution of the paper is extend the code length of EAQMDS in the literature (Guo et al. 2019). Consequently, the results show that almost all of these EAQMDS codes are new in the sense that the parameters of these codes are not covered by the previously known ones. [ABSTRACT FROM AUTHOR]

We consider the decoding of linear and array codes from errors when we are only allowed to download a part of the codeword. More specifically, suppose that we have encoded $k$ data symbols using an $(n,k)$ code with code length $n$ and dimension $k$. During storage, some of the codeword coordinates might be corrupted by errors. We aim to recover the original data by reading the corrupted codeword with a limit on the transmission bandwidth, namely, we can only download an $\alpha $ proportion of the corrupted codeword. For a given $\alpha $ , our objective is to design a code and a decoding scheme such that we can recover the original data from the largest possible number of errors. A naive scheme is to read $\alpha n$ coordinates of the codeword. This method used in conjunction with MDS codes guarantees recovery from any $\lfloor (\alpha n-k)/2\rfloor $ errors. In this paper we show that we can instead download an $\alpha $ proportion from each of the codeword’s coordinates. For a well-designed MDS code, this method can guarantee recovery from $\lfloor (n-k/\alpha)/2 \rfloor $ errors, which is $1/\alpha $ times more than the naive method, and is also the maximum number of errors that an $(n,k)$ code can correct by downloading only an $\alpha $ proportion of the codeword. We present two families of such optimal constructions and decoding schemes of which one is based on Interleaved Reed-Solomon codes and the other on Folded Reed-Solomon codes. We further show that both code constructions attain asymptotically optimal list decoding radius when downloading only a part of the corrupted codeword. We also construct an ensemble of random codes that with high probability approaches the upper bound on the number of correctable errors when the decoder downloads an $\alpha $ proportion of the corrupted codeword. [ABSTRACT FROM AUTHOR]

In this paper, we propose quantum stabilizer codes based on a new construction of self-orthogonal trace-inner product codes over the Galois field with 4 elements (GF(4)). First, from any two binary vectors, we construct a generator matrix of linear codes whose components are over GF(4). We prove that the proposed linear codes comply with the self-orthogonal, trace-inner product. Then, we propose mapping tables to construct new quantum stabilizer codes by using linear codes. Comparison results show that our proposed quantum codes have various dimensions for any code length with the capacity for better errors correction relative to the referenced quantum codes. [ABSTRACT FROM AUTHOR]

We present a new class of linear error-correcting codes taking numerical data into codewords with numerical symbols. These codes can correct large random numerical errors added to codewords. The goal is for numerical data to be protected directly as numbers whether in storage or transmission. The coding structures are based on discrete Fourier transform (DFT) codes, defined by parity-check matrices where rows are consecutively indexed DFT vectors. The new tensor codes use the Kronecker product of two parity-check DFT matrices of shorter length codes. We construct all necessary processing matrices. Error correction methods involve two levels of syndromes and use several stochastic Berlekamp-Massey Algorithms (sBMA) to find big syndromes, locate and evaluate large errors. Decoding is done in two stages with stage one producing corrected syndromes. Stage two determines errors within a well-defined segment for the codewords. Tensor codes have simpler and more efficient decoding operations. The characteristics of the tensor codes are compared to standard DFT codes of equal length. The processing operations are significantly less for the tensor codes but the standard DFT codes sometimes have better correcting performance. Nevertheless, tensor product codes can have acceptable levels of correction, more efficiently. [ABSTRACT FROM AUTHOR]

Linear block code (LBC) is an error detection and correction code that is widely used in communication systems. In this paper a special type of LBC called Hamming code was implemented and debugged using FPGA kit with integrated software environments ISE for simulation and tests the results of the hardware system. The implemented system has the ability to correct single bit error and detect two bits error. The data segments length was considered to give high reliability to the system and make an aggregation between the speed of processing and the hardware ability to be implemented. An adaptive length of input data has been consider, up to 248 bits of information can be handled using Spartan 3E500 with 43% as a maximum slices utilization. Input/output data buses in FPGA have been customized to meet the requirements where 34% of input/output resources have been used as maximum ratio. The overall hardware design can be considerable to give an optimum hardware size for the suitable information rate. [ABSTRACT FROM AUTHOR]

In this paper, we present linear systematic error-correcting codes Lk and L+k which are the results of our research on the sub-exceeding functions. Given an integer k such that k ≥ 3, these two codes are re-spectively [2k, k] and [3k, k] linear codes. The minimum distance of L3 is 3 and for k ≥ 4 the minimum distance of Lk is 4. The code L+k, the minimum distances are respectively 5 and 6 for k = 4 and k ≥ 5. By calculating the complexity of the algorithms, our codes have fast and efficient decoding. Then, for a short and medium distance data transmission (wifi network, bluetooth, cable, ...), we see that the codes mentioned above present many advantages. [ABSTRACT FROM AUTHOR]

We give an infinite family of maximum rank distance (MRD) codes, which covers properly the largest known linear MRD code family. Our family contains infinite families of non-linear MRD codes, which are the first non-linear examples for most of the parameters. We also give explicit examples and a table that demonstrates the proportion of linear and non-linear families for some small parameters. [ABSTRACT FROM PUBLISHER]

Two families of new asymmetric quantum codes are constructed in this paper. The first family is the asymmetric quantum codes with length $$n=q^{m}-1$$ over $$F_{q}$$ , where $$q\ge 5$$ is a prime power. The second one is the asymmetric quantum codes with length $$n=3^{m}-1$$ . These asymmetric quantum codes are derived from the CSS construction and pairs of nested BCH codes. Moreover, let the defining set $$T_{1}=T_{2}^{-q}$$ , then the real Z-distance of our asymmetric quantum codes are much larger than $$\delta _\mathrm{max}+1$$ , where $$\delta _\mathrm{max}$$ is the maximal designed distance of dual-containing narrow-sense BCH code, and the parameters presented here have better than the ones available in the literature. [ABSTRACT FROM AUTHOR]

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov–Tenengol’ts–Constantin–Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors. [ABSTRACT FROM AUTHOR]

Quantum synchronizable error-correcting codes are special quantum error-correcting codes that are designed to correct both the effect of quantum noise on qubits and misalignment in block synchronization. It is known that, in principle, such a code can be constructed through a combination of a classical linear code and its subcode if the two are both cyclic and dual-containing. However, finding such classical codes that lead to promising quantum synchronizable error-correcting codes is not a trivial task. In fact, although there are two families of classical codes that are proved to produce quantum synchronizable codes with good minimum distances and highest possible tolerance against misalignment, their code lengths have been restricted to primes and Mersenne numbers. In this paper, examining the incidence vectors of projective spaces over the finite fields of characteristic 2, we give quantum synchronizable codes from cyclic codes whose lengths are not primes or Mersenne numbers. These projective geometric codes achieve good performance in quantum error correction and possess the best possible ability to recover synchronization, thereby enriching the variety of good quantum synchronizable codes. We also extend the current knowledge of cyclic codes in classical coding theory by explicitly giving generator polynomials of the finite geometric codes and completely characterizing the minimum weight nonzero codewords. In addition to the codes based on projective spaces, we carry out a similar analysis on the well-known cyclic codes from Euclidean spaces that are known to be majority logic decodable and determine their exact minimum distances. [ABSTRACT FROM PUBLISHER]