The writing operation of multi-level flash memories can suffer from voltage overshoots, which can be generally modeled as asymmetric errors of limited magnitude. Using suitable error correcting codes, these kinds of errors can be corrected. In particular, q-ary non-linear codes of length 2 are equivalent to packings of the plane modulo q with quasi-crosses. The design procedures for a number of such packings are presented.

Linear covering codes and covering sets for the limited-magnitude-error channel are studied. Two new general covering set constructions are given.

In multilevel flash memories, leakage of charges results in errors, and the errors are asymmetric, of increasing type and of limited magnitude. On the other hand, low data retention may result in asymmetric errors of decreasing type and usually of smaller magnitude. Therefore, we have unbalanced error types. In this paper, some codes for correcting such errors are presented.

Binary linear [n,k] codes that are proper for error detection are known for many combinations ofnandk. For the remaining combinations, existence of proper codes is conjectured. In this paper, a particular class of [n,k] codes is studied in detail. In particular, it is shown that these codes are proper for many combinations ofnandkwhich were previously unsettled.

Upper bounds on the weight distribution function for codes of minimum distance at least 2 are given. Codes, where the bound is met with equality, are characterized. An improved upper bound on the weight distribution function for codes of minimum distance at least 3 is given. As an application, a sharp upper bound on the probability of undetected error for linear codes with full support is characterized.

An error model with symmetric errors of limited magnitude is considered. Limited magnitude means that the size of any error is limited by a number smaller (usually much smaller) than the alphabet size. Several constructions of codes correcting a single error are given. In some cases, the codes are perfect or quasi- perfect.

Lower bounds on the number of permutations p of {1, 2, . . . , n} satisfying |p i − i| ≤ d for all i are given.

A relatively new model of error correction is the limited magnitude error model. That is, it is assumed that the absolute difference between the sent and received symbols is bounded above by a certain value$l$. In this paper, we propose systematic codes for asymmetric limited magnitude channels that are able to correct a single error. We also show how this construction can be slightly modified to design codes that can correct a single symmetric error of limited magnitude. The designed codes achieve higher code rates than single error correcting codes previously given in the literature.

An error model with asymmetric errors of limited magnitude is a good model for some multilevel flash memories. This paper is about constructions of codes correcting such errors. The main results are about codes correcting a single such error and codes of lengthmcorrecting all errors inm-1 or less positions.

It is an open question if there exists proper binary linear codes for error detection for all lengths and dimensions. It is known that such codes exist for any given dimension when the length is above some explicit bound. The main result in this paper is new explicit bound that is much lower than the previously known bound. The construction is based on self-complementary codes. As an illustration, proper self-complementary codes are constructed for dimensions 5 and 6 and all lengths.

An(n,d) permutation array (PA) is a subset ofSnwith the property that the distance (under some metric) between any two permutations in the array is at leastd. They became popular recently for communication over power lines. Motivated by an application to flash memories, in this paper, the metric used is the Chebyshev metric. A number of different constructions are given, as well as bounds on the size of such PA.

Codes for error detection on aq-ary symmetric channel are studied. Whether a code is good or not for error detection (in the technical sense) depends on the structure of the code. For some combinations of the main parameters length, size, and minimum distance, all code are good and for some other combinations all are ugly (stronger than not good). The purpose of this paper is to give bounds on the parameters for codes that are not good for error detection. In particular, it is shown that if the minimum distance is below some bound, which depends on the length and size of the code as well as a lower bound on the number of codewords of minimum distance, then the code is ugly and, hence, not good for error detection.

It is shown that for any q and any size M, there exist proper codes for error detection on a q-ary symmetric channel for all sufficiently large lengths. The stronger condition zero-strong proper code is defined. It is shown that such codes can only exist for q dividing M, and if this is the case they are shown to exist for sufficiently large lengths.

It is shown that for any alphabet size$q$and any code size$M$, there exist proper codes for error detection on a$q$-ary symmetric channel for all sufficiently large lengths. The stronger conditionzero-strong propercode is defined. It is shown that such codes can only exist for$q$dividing$M$, and if this is the case they are shown to exist for sufficiently large lengths. The existence proofs are constructive.

Codes that can correct up totsymmetric errors and detect all unidirectional errors are studied. BOumlinck and van Tilborg gave a bound on the length of binary such codes. A generalization of this bound to arbitrary alphabet size is given. This generalized BOumlinck-van Tilborg bound, combined with constructions, is used to determine some optimal binary and ternary codes for correctingtsymmetric errors and detecting all unidirectional errors.

Computation of the undetected error probability for error detecting codes over the Z-channel is an important issue, explored only in part in previous literature. In this paper, Varshamov-Tenengol'ts (VT) codes are considered. First, an exact formula for the probability of undetected errors is given. It can be explicitly computed for small code lengths (up to approximately 25). Next, some lower bounds that can be explicitly computed up to almost twice this length are studied. A comparison to the Hamming codes is given. It is further shown that heuristic arguments give a very good approximation that can easily be computed even for large lengths. Finally, Monte Carlo methods are used to estimate performance for long code lengths.

Permutation arrays have found applications in powerline communication. One construction method for permutation arrays is to map good codes to permutations using a distance-preserving mappings (DPM). DPMs are mappings from the set of all q-ary vectors of a fixed length to the set of permutations of some fixed length (the same or longer) such that every two distinct vectors are mapped to permutations with the same or larger Hamming distance than that of the vectors. A DPM is called distance increasing (DIM) if the distances are strictly increased (except when the two vectors are equal). In this correspondence, we propose constructions of DPMs and DIMs from ternary vectors. The constructed DPMs and DIMs improve many lower bounds on the maximal size of permutation arrays.

Computation of the undetected error probability for error correcting codes over the Z-channel is an important issue, explored only in part in previous literature. In this paper we consider the case of Varshamov-Tenengol'ts codes, by presenting some analytical, numerical, and heuristic methods for unveiling this additional feature.

In this paper, a coding-theory construction of Cartesian authentication codes is presented. The construction is a generalization of some known constructions. Within the framework of this generic construction, several classes of authentication codes using certain classes of error-correcting codes are described. The authentication codes presented in this paper are better than known ones with comparable parameters. It is demonstrated that the construction is related to certain combinatorial designs, such as difference matrices and generalized Hadamard matrices

Bose and Lin introduced a class of systematic codes for detection of binary asymmetric errors. In this note, we describe a generalization to q-ary asymmetric error detecting codes. For these codes, the possible undetectable errors are characterized and the undetectable errors of minimum weight are determined

A new upper bound on the cardinality of codes in the Hamming space with given minimum and maximum distances is proved. The bound is compared to some known bounds, and some classes of codes for which the new bound is tight are given

Linear codes for error detection on a q-ary symmetric channel are studied. It is shown that for given dimension k and minimum distance d, there exists a value /spl mu/(d, k) such that if C is a code of length n /spl ges/ /spl mu/(d,k), then neither C nor its dual C/sup /spl perp// are good for error detection. For d /spl Gt/ k or k /spl Gt/ d good approximations for /spl mu/(d, k) are given. A generalization to nonlinear codes is also given.

The monotone structure of correctable and uncorrectable errors given by the complete decoding for a binary linear code is investigated. New bounds on the error-correction capability of linear codes beyond half the minimum distance are presented, both for the best codes and for arbitrary codes under some restrictions on their parameters. It is proved that some known codes of low rate are as good as the best codes in an asymptotic sense.

Corrupted packets that cause retransmission requests in automatic retransmission request (ARQ) systems can be reused. They can be combined with additional stored copies of the transmitted packet in order to obtain a single packet which is more reliable than any of the constituents. A scheme which suits the Z-channel is proposed here and the performance is analyzed under different coding assumptions.

Bose and Lin introduced a class of systematic codes for the detection of asymmetric errors (or equivalently, unidirectional errors). The determination of the probability of undetected error for these codes has been an open problem for many years. In this correspondence, the undetectable errors are characterized and the probability of undetected error is determined. Some detailed examples are given.

In this paper, the code that capable of correcting t-errors and detecting unidirectional errors (t-EC-AUED code) is presented. The t-EC AUED code is studied with the maximal size by using Sperner's theorem. The theorem says that the balanced code is an optimal all unidirectional error detecting code with the maximum number of code words.

Bose and Lin introduced a class of systematics codes for the detection of unidirectional errors (or equivalently, asymmetric errors). The codes are described, the undetectable errors are characterized, and the probability of undetected error for these codes is determined.

The maximum of g/sub 2/ - d/sub 2/ for linear [n,k,d;q] codes C is studied. Here d/sub 2/ is the smallest size of the support of a two-dimensional subcode of C and g/sub 2/ is the smallest size of the support of a two-dimensional subcode of C which contains a codeword of weight d. For codes of dimension 4 or more, upper and lower bounds on the maximum of g/sub 2/-d/sub 2/ are given.

We develop a connection between permutation arrays that are used in powerline communication and well-studied combinatorial objects, mutually orthogonal latin squares (MOLS). From this connection, many new results on permutation arrays can be obtained.

In this correspondence, two new constructions of permutation arrays are given. A number of examples to illustrate the constructions are also provided.

The probability of correct decoding on the binary-symmetric channel is studied. In particular, a class of codes with the same lengths and dimensions as the linear simplex codes, but with larger probability of correct decoding for all parameters p, 0 < p < 1/2, is given.

The number of cosets of weight 2/sup k-2/ or less are determined for the [2/sup k/-1, k, 2/sup k-1/] simplex code and a [2/sup k/-1, k, 2/sup k-1/-1] code obtained by a simple modification of the simplex code. The result proves that the [2/sup k/-1, k] simplex codes are not optimal for error correction on the binary symmetric channel with small bit error probability, p, (for k/spl ges/3). A proof that the modified code is better for all p, 0<p<1/2, is sketched.

Mappings of the set of binary vectors of a fixed length to the set of permutations of the same length are useful for the construction of permutation codes. In this article, several explicit constructions of such mappings preserving or increasing the Hamming distance are given. Some applications are given to illustrate the usefulness of the construction. In particular, a new lower bound on the maximal size of permutation arrays (PAs) is given.

In this paper, the undetected error probability for binary codes is studied. First complementary codes are studied. Next, a new proof of Abdel-Ghaffar's (1997) lower bound on the undetected error probability is presented and some generalizations are given. Further, upper and lower bounds on the undetected error probability for binary constant weight codes are given, and asymptotic versions are studied.

For a binary code C of length n, let C~ = V/sub n//spl bsol/C, be the complementary code. The main result of this paper is to determine K(n), the largest integer such that C~ is good for error detection for all linear [n, k] codes C with k/spl les/K(n).

A permutation array (PA) of length n and minimum distance d is a set of permutations of n elements such that any two permutations coincide in at most n - d positions. Some constructions of PAs are given.

Svanstrom (see IEEE ibid., vol.43, p.1630-2, Sept. 1997) gave a lower bound on the size of ternary constant-weight codes (CWCs). This bound is generalized and improved in some cases.

The well-known m-out-of-n code /spl Omega//sub n//sup m/ consists of all binary vectors of length n and weight m. It is known that it is good for error detection (in the technical sense, that is, the probability of undetected error P/sub ud/(/spl Omega//sub n//sup m/,p)/spl les/P/sub ud/(/spl Omega//sub n//sup m/,1/2) for all p, 0/spl les/p/spl les/1/2) only for a few small values of m and n. It is therefore of interest to determine (bounds for) the threshold in general, that is, find the range of bit-error probabilities p for which P/sub ud/ (/spl Omega//sub n//sup m/,p)/spl les/P/sub ud/ (/spl Omega//sub n//sup m/,1/2). In this article such bounds are given.

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d/sub 1/,d/sub 2/,/spl middot//spl middot//spl middot/,d/sub k/) where d/sub r/ is the smallest support of an r-dimensional subcode of C. An [n,k;q] code is extremal nonchain if, for any r and s, where 1/spl les/r

We study the Hamming distance d/sub H/(X,Y) between two independent identical distributed (i.i.d.) random n-tuples X and Y over some finite set, both lower and upper bounds are derived for the expectation Ed/sub H/(X,Y) and the variance Dd/sub H/(X,Y). Also, a generalization of the Grey-Rankin bound is given.

The Newton radius of a code is the largest weight of a uniquely correctable error. The covering radius is the largest distance between a vector and the code. Two relations between the Newton radius and the covering radius are given.

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d/sub 1/,d/sub 2/,...,d/sub k/) where d/sub r/ is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes. For each class bounds and extremal codes are discussed.

The error detection capability of almost-MDS (AMDS) and nearly-MDS (NMDS) codes is studied. Necessary and sufficient conditions for the codes to be proper or good for error detection are given.

For a binary linear code C of minimum distance d, if t>(d-1)/2, then there are errors of weight t which are not uniquely correctable. However, in many cases there are also errors of weight t which are uniquely correctable. The Newton radius of a code is defined to be the largest weight of a uniquely correctable error. Bounds and exact values of the Newton radius are given for several classes of codes.

The information function e/sub h/ of a code is the average amount of information contained in h positions of the codewords. Upper and lower bounds on the information function of binary linear codes are given. The average value and variance of the information function over all [n, k] codes are determined,.

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d/sub 1/,d/sub 2/,...d/sub k/) where d/sub /spl tau// is the smallest support of an /spl tau/-dimensional subcode of C. The possible weight hierarchies of [n,4;q] codes are studied. In particular, the possible weight hierarchies of [n,4;3] codes are determined.

Bounds on the weight hierarchies of the product of two simplex codes, two first order Reed-Muller codes, and the product of a simplex code and a first-order Reed-Muller code are determined. The weight hierarchies of the product of two Hamming codes and the product of a Hamming code and an even-weight code are also discussed.

The worst case probability of undetected error for a linear [n,k:q] code used on a local binomial channel is studied. For the two most important cases it is determined in terms of the weight hierarchy of the code. The worst case probability of undetected error is determined explicitly for some classes of codes.

The error detecting capability of Reed-Muller codes is analyzed. If a block code is used for error detection only, then an error is undetectable if it transforms a codeword into another codeword. The probability of undetected error for a binary [n,k] code C used on a binary-symmetric channel with crossover probability p, is given.

Upper and lower bounds on the weight distribution of a proper coset of a code are given.

The minimum support weight, d/sub r/(C), of a linear code C over GF(q) is the minimal size of the support of an r-dimensional subcode of C. A number of bounds on d/sub r/(C) are derived, generalizing the Plotkin bound and the Griesmer bound, as well as giving two new existential bounds. As the main result, it is shown that there exist codes of any given rate R whose ratio d/sub rd/sub 1/ is lower bounded by a number ranging from (q/sup r/-1)/(q/sup r/-q/sup r-1/) to r, depending on R.>

Codes of (d,k) sequences of constant Hamming weight are considered. Perfect codes correcting a single insertion, deletion, or peak-shift are defined and shown to exist. A systematic method to construct large classes of perfect codes is given. A number of related problems are considered.>

Upper and lower bounds on the average worst-case probability of undetected error for linear [n,k,q] codes are given.>

The authors considered weight hierarchies of codes satisfying the chain condition, they called these chain-good. First, they gave a set of simple necessary conditions for a sequence to be chain-good. They proved that given one chain-good sequence, there is an infinite set of chain-good sequences that can be constructed from this one sequence. Finally, they used this result to completely describe the sets of chain-good sequences of dimensions up to 5.>

Sullivan's (1967) inequality between the weight distribution function of a binary linear code and the weight distribution function of a proper subset of the code is generalized to linear codes over arbitrary finite fields.>

Some relations between the minimum support weights are discussed. In particular, the possible weight hierarchies of codes of dimension 4 are determined.>

The generalized Hamming weight, d/sub r/(C), of a binary linear code C is the size of the smallest support of any r-dimensional subcode of C. The parameter d/sub r/(C) determines the code's performance on the wire-tap channel of Type II. Bounds on d/sub r/(C), and in some cases exact expressions, are derived. In particular, a generalized Griesmer bound for d/sub r/(C) is presented and examples are given of codes meeting this bound with equality.>

The probability of undetected error for codes over GF(2/sup m/) having a generator matrix over GF(2) is studied. The optimal codes of dimensions four or less are determined. For higher dimensions, some properties of optimal codes are determined.>

It has been conjectured by T. Chang et al. (1990) that the number of cross-join pairs in a maximum length linear sequence equals (2/sup n-1/-1)(2/sup n-1/-2)/6. A maximum length linear sequence (an m-sequence) of length 2/sup n/-1 is a binary sequence which satisfies a linear recurrence whose characteristic polynomial is primitive of degree n. The number of primitive polynomials is given by phi (2/sup n/-1)/n, where phi is Euler's phi -function. A proof of the conjecture is given.>

An (I,J)-DDD is a set of I disjoint sets of distinct difference sets each having J elements. A number of constructions are given. Upper and lower bounds on the maximal element in a DDD (disjoint distinct difference) set are given. It is shown that regular DDD sets exist for I>or approximately=4J.>

The definition of difference triangle sets (DTSs) and is short survey of some of their important applications is given. A lower bound on the maximal element in any (I, J)-(DTS) is then given, and a construction of DTSs is described. Finally, tables of the best known lower and upper bounds on optimal DTSs are presented.>

Applications of difference triangle sets are briefly described. New lower and upper bounds on the size of optimal difference triangle sets are given.>

In data transmission and storage systems, combined error correction and detection procedures are often used to provide high reliability. This paper considers the use of separate concatenated codesCandDfor error correction and detection, respectively. It examines the error detection performance of codeDto determine how the probability of undetected error depends on the choice ofCandD. A comparison is made of the probability of undetected error achievable by codeDwith and without error correction, respectively. Asymptotic bounds for low bit error rates are developed which provide criteria to be considered when choosing codesCandD.

The probability of undetected error is studied when a code is used both for error correction and error detection. A number of generalizations is given of an upper bound of Korzhik on the minimal probability of undetected error for an(n, k)code.

A linear codeCover GF(q)is good fort-error-correction and error detection ifP(C,t;\epsilon) \leq P(C,t;(q - 1)/q)for all\epsilon, 0 \leq \epsilon \leq (q - 1)/q, whereP(C, t; \epsilon)is the probability of an undetected error after a codeword inCis transmitted over aq-ary symmetric channel with error probability\epsilonand correction is performed for all error patterns withtor fewer errors. A sufficient condition for a code to be good is derived. This sufficient condition is easy to check, and examples to illustrate the method are given.

We study the probability of having an undetected error when a linear block code is used to correct up toterrors on a symmetric channel, and the remaining power of the code is used for error detection.

The probability of undetected error of linear block codes for use on a binary symmetric channel is investigated. Upper hounds are derived. Several classes of linear block codes are proved to have good error-detecting capability.

A Survey is given of known upper bounds on codes correcting asymmetric errors. The bounds are improved by introducing new Ideas. By solving a linear programming problem an upper bound is given that is easy to compute for all codelengths and all minimum asymmetric distances.

Upper bounds on the covering radius of binary codes are studied. In particular it is shown that the covering radiusr_{m}of the first-order Reed-Muller code of lenglh2^{m}satisfies2^{m-l}-2^{\lceil m/2 \rceil -1} r_{m} \leq 2^{m-1}-2^{m/2-1}.