On the Hamming distance properties of group codes

Data stored in non-volatile memories may be destructed due to crosstalk and radiation but we can restore their data by using error-correcting codes. However, non-volatile memories consume a large amount of energy in writing. How to reduce writing bits even using error-correcting codes is one of the challenges in non-volatile memory design. In this paper, we propose a new write-reducing and error-correcting code, called Doughnut code. Doughnut code is based on state encoding limiting maximum and minimum Hamming distances. After that, we propose a code expansion method, which improves minimum and maximum Hamming distances by expanding a write-reducing code. When we apply our code expansion method to Doughnut code, we can obtain a write-reducing code whose error-correcting ability is equal to Hamming code. Experimental results show that the proposed write-reducing code reduces the number of writing bits by up to 36% compared to Hamming code.

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

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.

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.

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.

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.

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.

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>>

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.

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.

The permutation code (or the code) is well known object of research starting from 1970s. The code and its properties is used in different algorithmic domains such as error-correction, computer search, etc. It can be defined as follows: the set of permutations with the minimum distance between every pair of them. The considered distance can be different. In general, there are studied codes with Hamming, Ulam, Levensteins, etc. distances.In the paper we considered permutations codes over 2-Sylow subgroups of symmetric groups with Hamming distance over them. For this approach representation of permutations by rooted labeled binary trees is used. This representation was introduced in the previous author's paper. We also study the property of the Hamming distance defined on permutations from Sylow 2-subgroup $Syl_2(S_{2^n})$ of symmetric group $S_{2^n}$ and describe an algorithm for finding the Hamming distance over elements from Sylow 2-subgroup of the symmetric group with complexity $O(2^n)$. The metric properties of the codes that are defined on permutations from Sylow 2-subgroup $Syl_2(S_{2^n})$ of symmetric group $S_{2^n}$ are studied. The capacity and number of codes for the maximum and the minimum non-trivial distance over codes are characterized.

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].