A write-reducing and error-correcting code generation method for non-volatile memories

Non-volatile memories are paid attention to as a promising alternative to memory design. Data stored in them still may be destructed due to crosstalk and radiation. We can restore the data by using error-correcting codes which require extra bits to correct bit errors. Further, non-volatile memories consume ten to hundred times more energy than normal memories in bit-writing. When we configure them using error-correcting codes, it is quite necessary to reduce writing bits. In this paper, we propose a method to generate a bit-write-reducing code with error-correcting ability. We first pick up an error-correcting code which can correct t-bit errors. We cluster its codeswords and generate a cluster graph satisfying the S-bit flip conditions. We assign a data to be written to each cluster. In other words, we generate one-to-many mapping from each data to the codewords in the cluster. We prove that, if the cluster graph is a complete graph, every data in a memory cell can be re-written into another data by flipping at most S bits keeping error-correcting ability to t bits. We further propose an efficient method to cluster error-correcting codewords. Experimental results demonstrate that, when we apply our bit-write-reducing code to MediaBench applications, it can reduce writing-bit counts by up to 28.2% and also energy consumption of non-volatile memory cells by up to 27.9% compared to existing error-correcting codes keeping the same error-correcting ability. This paper proposes the world-first theoretically near-optimal bit-write-reducing code with error-correcting ability based on the efficient coding theories.

Linear error-correcting codes form the mathematical backbone of modern digital communication and storage systems, but identifying champion linear codes (linear codes achieving or exceeding the best known minimum Hamming distance) remains challenging. By training a transformer to predict the minimum Hamming distance of a class of linear codes and pairing it with a genetic algorithm over the search space, we develop a novel method for discovering champion codes. This model effectively reduces the search space of linear codes needed to achieve champion codes. Our results present the use of this method in the study and construction of error-correcting codes, applicable to codes such as generalised toric, Reed–Muller, Bose–Chaudhuri–Hocquenghem, algebrogeometric, and potentially quantum codes.

Error correction coding gain can be obtained with dc free magnetic recording codes having a minimum Hamming distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_{\min}\geq 4</tex> . In this information theoretic paper we establish lower bounds on the minimum Hamming distance achievable with (2n,k) (ie R = k/2n) block codes with runlength constrained or dc free sequences, and present a table with the computed guaranteed minimum values of k for (2n,k) dc free block codes with (b=0, l=2n-1, C=n) sequences, a specified block length 2n and desired minimum Hamming distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_{\min}</tex> . Subsequently, we present the formal synthesis of a (16,8) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_{\min}=4</tex> code with sequences which have, in NRZ notation, a minimum runlength of 1 and maximum runlength of 8, while the bound on the maximum instantaneous accumulated charge is C = 5. This accumulated charge returns to zero at the end of each 16 bit code word. Encoding and decoding schemes for the synthesized code are also proposed.

An [$$n, k, d$$n,k,d] code is a binary linear code of block length $$n$$n, dimension $$k$$k and minimum Hamming distance $$d$$d. Since the minimum distance determines the error detection or correction capability, it is desired that $$d$$d is as large as possible for the given block length $$n$$n and dimension $$k$$k. One of the most fundamental problems in coding theory is to construct codes with best possible minimum distances. This problem is very difficult in both theory and practice. During the last decades, it has proved that the class of quasi-cyclic (QC) codes contain many such codes. In this paper, augmentation of binary QC codes is studied. A new augmentation algorithm is presented, and 10 new $$h$$h-generator QC codes that are better than previously known code have been constructed. Furthermore, Construction X has been applied to obtain another 18 new improved binary linear codes. With the standard construction techniques, a total of 124 new binary linear codes that improve the lower bound on the minimum distance have been obtained.

Runlength-limited (RLL) and digital-sum-limited (DSL) codes are considered. For these codes the finite and asymptotic lower bounds on achievable rates for the given minimum Hamming distance are derived. Using generating functions and trellis diagram techniques, the authors prove the existence of RLL- and DSL-codes of rate R and minimum distance d= delta n, such that R>or=2.log lambda /sub 1/(0)-min(log lambda /sub 1/(z)- delta .log z), 0<or=z<or=1, for large enough code length n. The value lambda /sub 1/(z) is the maximum modulus eigenvalue of a certain matrix that is dependent on the limitations and some parameter z. The bounds for the limited codes improve the lower bound of H. C. Ferreira (1984).<<ETX>>

Techniques are presented for computing upper and lower bounds on the number of errors that can be corrected by list decoders for general block codes and, specifically, for Reed--Solomon (RS) codes. The list decoder of Guruswami and Sudan implies such a lower bound (referred to here as the GS bound) for RS codes. It is shown that this lower bound, given by means of the code's length, the minimum Hamming distance, and the maximal allowed list size, in fact applies to all block codes. Ranges of code parameters are identified where the GS bound is tight for worst-case RS codes, in which case the list decoder of Guruswami and Sudan provably corrects the largest possible number of errors. On the other hand, ranges of parameters are provided for which the GS lower bound can be strictly improved. In some cases the improvement applies to all block codes with a given minimum Hamming distance, while in others it applies only to RS codes.

Minimal Residual Disease (MRD) Status Predicts Outcomes in Patients with Follicular Lymphoma (FL) Treated with Chemo-Immunotherapy on SWOG S0016

Let [n, k, d]/sub q/-codes be linear codes of length n, dimension k, and minimum Hamming distance d over GF(q). In this article, 31 new quaternary (q=4) codes are constructed which improve the respective lower bounds on minimum distance in Brouwer's tables.

Under certain mild conditions, the minimum Hamming distance D of an (N, K, D) group code C over a non-abelian group G is bounded by D N/2. Consequently, there exists no (N, K, N-K+1) group code C over an non-abelian group G if 1 >

The symmetric group $$S_n$$ S n on $$n$$ n letters is a metric space with respect to the Hamming distance. The corresponding isometry group is well known to be isomorphic to the wreath product $$S_n \wr S_2$$ S n ? S 2 . A subset of $$S_n$$ S n is called a permutation code or a permutation array, and the largest possible size of a permutation code with minimum Hamming distance $$d$$ d is denoted by $$M(n, d)$$ M ( n , d ) . Using exhaustive search by computer on sets of orbits of isometry subgroups $$U$$ U we are able to determine serveral new lower bounds for $$M(n,d)$$ M ( n , d ) for $$n \le 22$$ n ≤ 22 . The codes are given by the group $$U$$ U and representatives of the $$U$$ U -orbits.

Classical Turbo code suffers from high error floor due to its small minimum Hamming distance (MHD). Newly-proposed 3D-Turbo code can effectively increase the MHD and achieve a lower error floor by adding a rate-1 post encoder, through which part of the parity bits from the classical Turbo encoder are further encoded. In this paper, a novel bit-interleaved coded modulation (BICM) system is proposed by combining rotated mapping QAM and 3D-Turbo code to effectively improve the performance of 3D-Turbo code over Raleigh fading channels. A 2D iterative soft demodulating-decoding algorithm is developed for the proposed BICM system. Simulation results show that the proposed system can obtain about 0.8-1.0 dB gain at bit error rate (BER) of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> , compared with the existing BICM system with Gray mapping QAM.

This work addresses the problem of increasing the robustness to bit errors for two description scalar quantizers. To this aim a permutation is applied to the indices of each description. We show how to construct permutation pairs that increase the minimum Hamming distance of the set of valid index pairs to 3 when the redundancy is sufficiently high. Additionally, for the case when one description is known to be correct we propose a new performance criterion, denoted by d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">side,min</sub> . This represents the minimum Hamming distance of the set of valid indices of one description, when the index of the other description is fixed. We develop a technique for constructing permutation pairs achieving d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">side</sub> , <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> ≥ h based on a linear (R, [log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> m]) channel code of minimum Hamming distance h + 1, where R is the rate of each description and m is the number of diagonals occupied by the valid index pairs in the matrix of the initial index assignment.

Run-length limited (RLL) codes can facilitate reliable data transmission and provide flicker-free illumination in visible light communication (VLC) systems. We propose novel high-rate RLL codes, which can improve error performance and mitigate flicker. Two RLL coding schemes are developed by designing the finite-state machine to further enhance the coding gain by improving the minimum Hamming distance and using the state-splitting method to realize small state numbers. In our RLL code design, the construction of the codeword set is critical. This codeword set is designed considering the set-partitioning algorithm criterion. The flicker control and minimum Hamming distance of the various proposed RLL codes are described in detail, and the flicker performances of different codes are compared based on histograms. Simulations are conducted to evaluate the proposed RLL codes in on-off keying modulation VLC systems. Simulation results demonstrate that the proposed RLL codes achieve superior error performance to the existing RLL codes.

Let C be an [n, k, d] binary linear block code of length n, dimension k and minimum Hamming distance d over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ). Let d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sup> denote the minimum Hamming distance of the dual code of C over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ). Let ¿ : GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) ¿ {-1, 1}n be the component-wise mapping ¿(v <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ) := (-1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vi</sup> , for v =(v <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , v <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , ... , v <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) ¿ GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ). Finally, for p < n, let ¿ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</sub> be a p × n random matrix whose rows are obtained by mapping a uniformly drawn set of size p of the codewords of C under e. Recently, the authors have established that for d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sup> large enough and y := p/n ¿ (0, 1) fixed, as n ¿ ¿ the empirical eigen-distribution of the Gram matrix of 1/¿n times ¿ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</sub> resembles that of a random i.i.d. Rademacher matrix (i.e., the Marchenko-Pastur distribution). In this paper, we overview this result and discuss its implications on the design of frames for compressed sensing applications.

Let [Formula: see text] be a finite field of order [Formula: see text] and [Formula: see text] be a prime such that [Formula: see text] where [Formula: see text] are distinct odd primes with [Formula: see text] and [Formula: see text] is a positive integer with [Formula: see text]. In 2022, Rakphon et al. obtained all irreducible divisors of [Formula: see text] over [Formula: see text] where [Formula: see text] and [Formula: see text]. In this paper, we determine all primitive idempotents which can generate [Formula: see text]-constacyclic codes in [Formula: see text] where parity-check polynomials of these codes are shown in 2022. Moreover, we find the minimum Hammimg distances of the codes which are generated by primitive idempotents when [Formula: see text].