New lower bounds on q-ary error-correcting codes

Constructing error-correcting binary codes using transitive permutation groups

In the convolutional code, the minimum free distance is the important parameter to evaluate the performance of the code. Recently, it has been considered necessary, however, for a more strict evaluation of the performance to examine the number of code words with weights around the minimum free distance, i.e., the more general distance spectrum. In general, it is difficult to derive the distance spectrum of a code, including the derivation of the minimum free distance when the code size is enalrged, and it will be significant if it is derived easily.This paper proposes an algorithm that can derive the distance spectrum with a high speed, compared to the method already proposed. The algorithm presented in this paper is an improvement of the fast algorithm for searching trees (FAST) proposed by M. Cedervall and R. Johannesson. The distance spectrum can be derived with approximately one‐half the computational complexity compared to FAST.

DOI: 10.1049/cp:20061337 ISBN: 0 86341 644 6 Location: Hangzhou, China Conference date: 6-9 Nov. 2006 Format: PDF The BER performance of presented quasi-cyclic (QC) low-density parity-check (LDPC) codes is not good as that of randomly constructed LDPC codes, while the lack of structure of randomly constructed LDPC codes implies serious disadvantages in terms of storing and accessing a large parity-check matrix. To solve the problems, this paper proposes a design of good quasi-cyclic (QC) low-density parity-check (LDPC) codes, the obtained algebraically structured codes have large minimum distances and good BER performance. The proposed design is based on index matrices, which determines the shifts of the circulant matrices of the sparse parity-check matrices. Compared with the presented design of QC LDPC codes, the designed QC LDPC codes are free from girth 4, sometimes free from girth 6, and have much larger minimum distances. Based on the algebraic code structure, the conditions of the girth and minimum distance of the codes are found. The BER performance of the designed QC LDPC block codes compares with that of randomly constructed LDPC codes for any block lengths. Better BER performance is obtained by increasing the circulant size of the base QC code. Simulations verify the construction of the QC LDPC codes to be valid. (4 pages) Inspec keywords: AWGN channels; error statistics; channel coding; algebraic codes; parity check codes; sparse matrices Subjects: Algebra; Other topics in statistics; Codes

Polarization-adjusted convolutional (PAC) codes of short blocklengths over the binary-input additive white Gaussian noise channel have excellent performance. To further improve the performance of the PAC codes, we adopt the asymptotic frame error rate (AFER) optimality criterion, i.e., the primary and secondary criteria are to maximize the minimum Hamming distance and to minimize the error coefficient of the code, respectively.In this paper, we conduct a new optimization search using a graphic processing unit that jointly optimizes the rate profile and the convolutional transform of a PAC code. Our search has discovered many short record-breaking binary linear block codes (representable as PAC codes) which achieve smaller error coefficients at the same best-known minimum distance compared to the best records in the literature. In particular, at a frame error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−9</sup> , an optimized [64, 32] PAC code with a minimum distance of 12 outperforms the best-known counterpart by 0.40 dB and is 1.1 dB away from the Polyanskiy-Poor-Verdú bound.

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.

A coordinate of a binary code of size M is said to be balanced if the number of zero and ones in the coordinate is either \(\lfloor M/2\rfloor \) or \(\lceil M/2 \rceil \) (that is, exactly \(M/2\) for even M). Since good codes (of various types) tend to be balanced in all coordinates, various conjectures have been made regarding the existence of such codes. It is here shown that there are parameters for which there are no optimal binary error-correcting codes with a balanced coordinate. This is proved by the code attaining \(A(17,8) = 36\), which is shown to be unique here; \(A(n,d)\) denotes the maximum size of a binary code of length n and minimum distance d. It is further shown that \(A(18,8) \le 68\).

The problem of finding the values of A_q(n,d)—the maximum size of a code of length n and minimum distance d over an alphabet of q elements—is considered. Upper and lower bounds on A_4(n,d) are presented and some values of this function are settled. A table of best known bounds on A_4(n,d) is given for n \leq 12. When q \leq M < 2q, all parameters for which A_q(n,d) = M are determined.

The maximum size of unrestricted binary three-error-correcting codes has been known up to the length of the binary Golay code, with two exceptions. Specifically, denoting the maximum size of an unrestricted binary code of length n and minimum distance d by A(n, d), it has been known that $$64 \le A(18,8) \le 68$$ and $$128 \le A(19,8) \le 131$$ . In the current computer-aided study, it is shown that $$A(18,8)=64$$ and $$A(19,8)=128$$ , so an optimal code is obtained even after shortening the extended binary Golay code six times.

The process of DNA-based data storage (DNA storage for short) can be mathematically modelled as a communication channel, termed DNA storage channel, whose inputs and outputs are sets of unordered sequences. To design error correcting codes for DNA storage channel, a new metric, termed the sequence-subset distance , is introduced, which generalizes the Hamming distance to a distance function defined between any two sets of unordered vectors and helps to establish a uniform framework to design error correcting codes for DNA storage channel. We further introduce a family of error correcting codes, referred to as sequence-subset codes , for DNA storage and show that the error-correcting ability of such codes is completely determined by their minimum distance. We derive some upper bounds on the size of the sequence-subset codes including a tight bound for a special case, a Singleton-like bound and a Plotkin-like bound. We also propose some constructions, including an optimal construction for that special case, which imply lower bounds on the size of such 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>>

In this paper, we study the behavior of the function $$t_q(k)$$ defined as the maximal deviation from the Griesmer bound of the optimal length of a linear code with a fixed dimension k: $$\begin{aligned} t_q(k)=\max _d(n_q(k,d)-g_q(k,d)), \end{aligned}$$ where the maximum is taken over all minimum distances d. Here $$n_q(k,d)$$ is the shortest length of a q-ary linear code of dimension k and minimum distance d, $$g_q(k,d)$$ is the Griesmer bound for a code of dimension k and minimum distance d. We give two equivalent geometric versions of this problem in terms of arcs and minihypers. We prove that $$t_q(k)\rightarrow \infty $$ when $$k\rightarrow \infty $$ which implies that the problem is non-trivial. We prove upper bounds on the function $$t_q(k)$$ . For codes of even dimension k we show that $$t_q(k)\le 2(q^{k/2}-1)/(q-1)-(k+q-1)$$ which implies that $$t_q(k)\in O(q^{k/2})$$ for all k. For three-dimensional codes and q even we prove the stronger estimate $$t_q(3)\le \log q-1$$ , as well as $$t_q(3)\le \sqrt{q}-1$$ for the case q square.

This paper studies the cardinality of codes correcting insertions and deletions. We give improved upper and lower bounds on code size. Our upper bound is obtained by utilizing the asymmetric property of list decoding for insertions and deletions and can be seen as analogous to the Elias bound in the Hamming metric. Our non-asymptotic bound is better than the existing bounds when the minimum Levenshtein distance is relatively large. The asymptotic bound exceeds the Elias and the MRRW bounds adapted from the Hamming-metric bounds for the binary and the quaternary cases. Our lower bound improves on the bound by Levenshtein, but its effect is limited and vanishes asymptotically.

One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate $R$ and minimum distance $\delta $ of a binary code. The best known tradeoff is the Gilbert–Varshamov bound, and says that for every $\delta \in (0, 1/2)$ , there are codes with minimum distance $\delta $ and rate $R = {R_{\mathsf {GV}}}(\delta) > 0$ (for a certain simple function ${R_{\mathsf {GV}}}(\cdot)$ ). In this paper, we show that the Gilbert–Varshamov bound can be achieved by codes, which support local error-detection and error-correction algorithms. Specifically, we show the following results. 1) Local testing: for all $\delta \in (0,1/2)$ and all $R , there exist codes with length $n$ , rate $R$ , and minimum distance $\delta $ that are locally testable with $\mathsf {quasipolylog}(n)$ query complexity. 2) Local correction: for all $\epsilon > 0$ , for all $\delta sufficiently large, and all $R , there exist codes with length $n$ , rate $R$ , and minimum distance $\delta $ that are locally correctable from $({\delta }/{2}) - o(1)$ fraction errors with $O(n^{\epsilon })$ query complexity. Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters. Our local testing result is obtained by combining Thommesen’s random concatenation technique and the best known locally testable codes by Kopparty et al. Our local correction result, which is significantly more involved, also uses random concatenation, along with a number of further ideas: the Guruswami–Sudan–Indyk list decoding strategy for concatenated codes, Alon–Edmonds–Luby distance amplification, and the local list-decodability, local list-recoverability, and local testability of Reed–Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing algorithms; this seems to be the first time local testability is used in the construction of a locally correctable code.

DNA codes consisting of DNA sequences are necessary for DNA computing. The minimum distance parameter of such codes is a measure of how dissimilar the codewords are, and thus is indirectly a measure of the likelihood of undetectedable or uncorrectable errors occurring during hybridization. To compute distance, an abstract metric, for example, longest common subsequence, must be used to model the actual bonding energies of DNA strands. In this paper we continue the development [1,2,3] of similarity functions for q-ary n-sequences The theoretical lower bound on the maximal possible size of codes, built on the space endowed with this metric, is obtained. that can be used (for q = 4) to model a thermodynamic similarity on DNA sequences. We introduce the concept of a stem similarity function and discuss DNA codes [2] based on the stem similarity. We suggest an optimal construction [2] and obtain random coding bounds on the maximum size and rate for such codes.KeywordsArbitrary IntegerRandom CodeUnique RootReverse ComplementLonge Common SubsequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.