The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect binary one-error-correcting code from its minimum distance graph is presented. Consequently, inequivalent such codes have nonisomorphic minimum distance graphs. Moreover, it is shown that the automorphism group of a minimum distance graph is isomorphic to that of the corresponding code.
One peculiarity with deletion-correcting codes is that perfect $t$-deletion-correcting codes of the same length over the same alphabet can have different numbers of codewords, because the balls of radius $t$ with respect to the Levenshte\uın distance may be of different sizes. There is interest, therefore, in determining all possible sizes of a perfect $t$-deletion-correcting code, given the length $n$ and the alphabet size~$q$. In this paper, we determine completely the spectrum of possible sizes for perfect $q$-ary 1-deletion-correcting codes of length three for all $q$, and perfect $q$-ary …
Motivated by applications to DNA-storage, flash memory, and magnetic recording, we study perfect burst-correcting codes for the limited-magnitude error channel. These codes are lattices that tile the integer grid with the appropriate error ball. We construct two classes of such perfect codes correcting a single burst of length $2$ for $(1,0)$-limited-magnitude errors, both for cyclic and non-cyclic bursts. We also present a generic construction that requires a primitive element in a finite field with specific properties. We then show that in various parameter regimes such primitive elements ex…
The doubly shortened perfect codes of length 13 are classified utilizing the classification of perfect codes in [P.R.J. Östergård and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I - Classification, IEEE Trans. Inform. Theory, to appear]; there are 117821 such (13,512,3) codes. By applying a switching operation to those codes, two more (13,512,3) codes are obtained, which are then not doubly shortened perfect codes.
A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 is presented. There are 5983 such inequivalent perfect codes and 2165 extended perfect codes. Efficient generation of these codes relies on the recent classification of Steiner quadruple systems of order 16. Utilizing a result of Blackmore, the optimal binary one-error-correcting codes of length 14 and the (15, 1024, 4) codes are also classified; there are 38408 and 5983 such codes, respectively.
A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. Östergård and O. Pottonen, "The perfect binary one-error-correcting codes of length 15: Part I--Classification," IEEE Trans. Inform. Theory vol. 55, pp. 4657--4660, 2009]. In the current accompanying work, the classified codes are studied in great detail, and their main properties are tabulated. The results include the fact that 33 of the 80 Steiner triple systems of order 15 occur in such codes. Further understanding is gained o…
We prove that every $1$-error-correcting code over a finite field can be embedded in a $1$-perfect code of some larger length. Embedding in this context means that the original code is a subcode of the resulting $1$-perfect code and can be obtained from it by repeated shortening. Further, we generalize the results to partitions: every partition of the Hamming space into $1$-error-correcting codes can be embedded in a partition of a space of some larger dimension into $1$-perfect codes. For the partitions, the embedding length is close to the theoretical bound for the general case and optimal f…
The Golomb-Welch conjecture states that there are no perfect $e$-error-correcting Lee codes in $\mathbb{Z}^n$ ($PL(n,e)$-codes) whenever $n\geq 3$ and $e\geq 2$. A special case of this conjecture is when $e=2$. In a recent paper of A. Campello, S. Costa and the author of this paper, it is proved that the set $\mathcal{N}$ of dimensions $n\geq 3$ for which there are no linear $PL(n,2)$-codes is infinite and $\#\{n \in \mathcal{N}: n\leq x\} \geq \frac{x}{3\ln(x)/2} (1+o(1))$. In this paper we present a simple and elementary argument which allows to improve the above result to $\#\{n \in \mathca…
The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by this combining construction is generalized to the $q$-ary case. Simply, every non-full-rank perfect code $C$ is the union of a well-defined family of $μ$-components $K_μ$, where $μ$ belongs to an "outer" perfect code $C^*$, and these components are at distance three from each other. Components from distinct codes can thus freely be combined to obtain new perfect codes. The Phelps general produc…
Chulalongkorn University（朱拉隆功大学）；Walailak University（瓦莱岚大学）；National Institute of Informatics（国立信息学研究所）
Multi-class classification is mandatory for real world problems and one of promising techniques for multi-class classification is Error Correcting Output Code. We propose a method for constructing the Error Correcting Output Code to obtain the suitable combination of positive and negative classes encoded to represent binary classifiers. The minimum weight perfect matching algorithm is applied to find the optimal pairs of subset of classes by using the generalization performance as a weighting criterion. Based on our method, each subset of classes with positive and negative labels is appropriat…
The class of poset metrics is very large and contains some interesting families of metrics. A family of metrics, based on posets which are formed from disjoint chains which have the same size, is examined. A necessary and sufficient condition, for the existence of perfect single-error-correcting codes for such poset metrics, is proved.
Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence and enumeration of diameter perfect Lee codes. As main results we determine all $q$ for which there exists a linear diameter-4 perfect Lee code of word length $n$ over $Z_{q},$ and prove that for each $n\geq 3$ there are uncountable many diameter-4 perfect Lee codes of word length $n$ over $Z.$ This is in a strict contrast with perfect error-correcting Lee co…
The rank modulation scheme has been proposed for efficient writing and storing data in non-volatile memory storage. Error-correction in the rank modulation scheme is done by considering permutation codes. In this paper we consider codes in the set of all permutations on $n$ elements, $S_n$, using the Kendall's $τ$-metric. We prove that there are no perfect single-error-correcting codes in $S_n$, where $n>4$ is a prime or $4\leq n\leq 10$. We also prove that if such a code exists for $n$ which is not a prime then the code should have some uniform structure. We define some variations of the Kend…
Permutation and multipermutation codes in the Ulam metric have been suggested for use in non-volatile memory storage systems such as flash memory devices. In this paper we introduce a new method to calculate permutation sphere sizes in the Ulam metric using Young Tableaux and prove the non-existence of non-trivial perfect permutation codes in the Ulam metric. We then extend the study to multipermutations, providing tight upper and lower bounds on multipermutation Ulam sphere sizes and resulting upper and lower bounds on the maximal size of multipermutation codes in the Ulam metric.
Several new applications and a number of new mathematical techniques have increased the research on error-correcting codes in the Lee metric in the last decade. In this work we consider several coding problems and constructions of error-correcting codes in the Lee metric. First, we consider constructions of dense error-correcting codes in relatively small dimensions over small alphabets. The second problem we solve is construction of diametric perfect codes with minimum distance four. We will construct such codes over various lengths and alphabet sizes. The third problem is to transfer an n-di…
The paper deals with the perfect 1-error correcting codes over a finite field with $q$ elements (briefly $q$-ary 1-perfect codes). We show that the orthogonal code to the $q$-ary non-full-rank 1-perfect code of length $n = (q^{m}-1)/(q-1)$ is a $q$-ary constant-weight code with Hamming weight equals to $q^{m - 1}$ where $m$ is any natural number not less than two. We derive necessary and sufficient conditions for $q$-ary 1-perfect codes of non-full rank. We suggest a generalization of the concatenation construction to the $q$-ary case and construct the ternary 1-perfect codes of length 13 and …
Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length $2^m-4$ and $2^m-3$, respectively) are optimal. Properties of such codes are here studied, determining among other things parameters of certain subcodes. A utilization of these properties makes a computer-aided classification of the optimal binary one-error-correcting codes of lengths 12 and 13 possible; there are 237610 and 117823 such codes, respectively (with 27375 and 17513 inequivalent extensions). This completes the classification of optimal binar…
The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. Error-correcting codes are essential for rank modulation, however, existing results have been limited. In this work we explore a new approach, \emph{systematic error-correcting codes for rank modulation}. Systematic codes have the benefits of enabling efficient information retrieval and potentially supporting more efficient encoding and decoding procedures. We study systematic codes for rank modulation under Kendall's $τ$-metric as well as under the $\ell_\infty$-metric. In Kendall's $τ$…
We study whether an asymmetric limited-magnitude ball may tile $\mathbb{Z}^n$. This ball generalizes previously studied shapes: crosses, semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting codes in a channel which changes a transmitted integer vector in a bounded number of entries by limited-magnitude errors. A construction of lattice tilings based on perfect codes in the Hamming metric is given. Several non-existence results are proved, both for general tilings, and lattice tilings. A complete classification of lattice tilings for two certain cases is proved.
We improve the upper bound of Levenshtein for the cardinality of a code of length 4 capable of correcting single deletions over an alphabet of even size. We also illustrate that the new upper bound is sharp. Furthermore we will construct an optimal perfect code capable of correcting single deletions for the same parameters.
In the rank modulation scheme for flash memories, permutation codes have been studied. In this paper, we study perfect permutation codes in $S_n$, the set of all permutations on $n$ elements, under the Kendall τ-Metric. We answer one open problem proposed by Buzaglo and Etzion. That is, proving the nonexistence of perfect codes in $S_n$, under the Kendall τ-metric, for more values of $n$. Specifically, we present the recursive formulas for the size of a ball with radius $r$ in $S_n$ under the Kendall τ-metric. Further, We prove that there are no perfect $t$-error-correcting codes in $S_n$ unde…
The rank modulation scheme has been proposed for efficient writing and storing data in non-volatile memory storage. Error-correction in the rank modulation scheme is done by considering permutation codes. In this paper we consider codes in the set of all permutations on $n$ elements, $S_n$, using the Kendall $τ$-metric. The main goal of this paper is to derive new bounds on the size of such codes. For this purpose we also consider perfect codes, diameter perfect codes, and the size of optimal anticodes in the Kendall $τ$-metric, structures which have their own considerable interest. We prove t…
We improve Levenshtein's upper bound for the cardinality of a code of length four that is capable of correcting single deletions over an alphabet of even size. We also illustrate that the new upper bound is sharp. Furthermore we construct an optimal perfect code that is capable of correcting single deletions for the same parameters.
Existence and Constructions of Strict Function-Correcting Codes with Data Protection
The International Institute of Information Technology Hyderabad（印度国际信息技术学院海得拉巴分校）；Indian Institute of Science, Bengaluru（印度科学研究所班加罗尔）；Aalto University（阿尔托大学）
An $n$-dimensional chair consists of an $n$-dimensional box from which a smaller $n$-dimensional box is removed. A tiling of an $n$-dimensional chair has two nice applications in coding for write-once memories. The first one is in the design of codes which correct asymmetric errors with limited-magnitude. The second one is in the design of $n$ cells $q$-ary write-once memory codes. We show an equivalence between the design of a tiling with an integer lattice and the design of a tiling from a generalization of splitting (or of Sidon sequences). A tiling of an $n$-dimensional chair can define a …
The Golomb-Welch conjecture deals with the existence of perfect $e$% -error correcting Lee codes of word length $n,$ $PL(n,e)$ codes. Although there are many papers on the topic, the conjecture is still far from being solved. In this paper we initiate the study of an invariant connected to abelian groups that enables us to reformulate the conjecture, and then to prove the non-existence of linear PL(n,2) codes for $n\leq 12$. Using this new approach we also construct the first quasi-perfect Lee codes for dimension $n=3,$ and show that, for fixed $n$, there are only finitely many such codes over…
In the current thesis we first talk about the six-qubit quantum error-correcting code and show its connections to entanglement-assisted error-correcting coding theory and then to subsystem codes. This code bridges the gap between the five-qubit (perfect) and Steane codes. We discuss two methods to encode one qubit into six physical qubits. Each of the two examples corrects an arbitrary single-qubit error. The first example is a degenerate six-qubit quantum error-correcting code. We prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Stean…
Mixed codes, which are error-correcting codes in the Cartesian product of different-sized spaces, model degrading storage systems well. While such codes have previously been studied for their algebraic properties (e.g., existence of perfect codes) or in the case of unbounded alphabet sizes, we focus on the case of finite alphabets, and generalize the Gilbert-Varshamov, sphere-packing, Elias-Bassalygo, and first linear programming bounds to that setting. In the latter case, our proof is also the first for the non-symmetric mono-alphabetic $q$-ary case using Navon and Samorodnitsky's Fourier-ana…
We consider lattice tilings of $\R^n$ by a shape we call a $(\kp,\km,n)$-quasi-cross. Such lattices form perfect error-correcting codes which correct a single limited-magnitude error with prescribed maximal-magnitudes of positive error and negative error (the ratio of which is called the balance ratio). These codes can be used to correct both disturb and retention errors in flash memories, which are characterized by having limited magnitudes and different signs. We construct infinite families of perfect codes for any rational balance ratio, and provide a specific construction for $(2,1,n)$-qua…
We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist. We derive an existence bound that is the equivalent of the Gilbert--Varshamov bound in Hamming metric. We study the asymptotic behavior of the minimum rank distance of codes satisfying GV. We derive the probability distribution of minimum rank distance for random and random $\F{q}$-linear codes. We give an asymptotic equivalent of their average minimum rank distance and show that random $\F{q}$-linear codes are on GV bound for rank metric. We show that the cov…
Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. Besides a global level, they enable errors to be corrected locally, reducing the need for communication between storage nodes. There is a close connection between almost affine LRCs and matroid theory which can be utilized to construct good LRCs and derive bounds on their performance. A generalized Singleton bound for linear LRCs with parameters $(n,k,d,r,δ)$ was given in [N. Prakash et al., "Optimal Linear Codes with a Local-Error-Correction Property", IEEE Int. Symp. Inf. Theory]. In this paper, a LR…
Recently, error correcting codes in the erasure channel have drawn great attention for various applications such as distributed storage systems and wireless sensor networks, but many of their decoding algorithms are not practical because they have higher decoding complexity and longer delay. Thus, the automorphism group decoder (AGD) for cyclic codes in the erasure channel was introduced, which has good erasure decoding performance with low decoding complexity. In this paper, we propose new two-stage AGDs (TS-AGDs) for cyclic codes in the erasure channel by modifying the parity check matrix an…
