a lower bound on the undetected error probability and strictly optimal codes error detection is a simple technique used in various communication and memory systems to enhance reliability we study the probability that a q ary linear or nonlinear block code of length n and size m fails to detect an error a lower bound on this undetected error probability is derived in terms of q n and m the new bound improves upon other bounds mentioned in the literature even those that hold only for linear codes block codes whose undetected error probability equals the new lower bound are investigated we call these codes strictly optimal codes and give a combinatorial characterization of them we also present necessary and sufficient conditions for their existence in particular we find all values of n and m for which strictly optimal binary codes exist and determine the structure of all of them for example we construct strictly optimal binary coded decimal codes of length four and five and we show that these are the only possible lengths of such codes

computing the nucleolus of cyclic permutation games abstract the question of allocating costs or benefits among the participants of a joint enterprise is frequently answered by determining the nucleolus of a related cooperative game we discuss this issue in connection with multi person decision situations in which finding the best course of joint action for a group of participants can be modeled by the well known assignment optimization problem the related transferable utility cooperative games are called permutation games we consider the large subclass of so called cyclic permutation games and show how the nucleolus of an n player cyclic permutation game can be computed by a known o n 4 algorithm directly from the underlying data we also demonstrate that this approach might not work if for the group of all participants the optimal course of action is not given by a permutation consisting of a single cycle

on the undetected error probability of triple error correcting bch codes the probability of undetected error p sub u epsilon for the primitive triple error correcting bch codes of blocklength 2 sup m 1 used solely for error detection on a binary symmetric channel with crossover probability epsilon

optimal codes for error detection 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

new upper bounds on error exponents we derive new upper bounds on the error exponents for the maximum likelihood decoding and error detecting in the binary symmetric channels this is an improvement on the best earlier known bounds by shannon gallager berlekamp 1967 and mceliece omura 1977 for the probability of undetected error the new bounds are better than the bounds by levenshtein 1978 1989 and the bound by abdel ghaffar see ibid vol 43 p 1489 502 1997 moreover we further extend the range of rates where the undetected error exponent is known to be exact the new bounds are based on an analysis of possible distance distributions of the codes along with some inequalities relating the distance distributions to the error probabilities

optimum cycle redundancy check codes with 16 bit redundancy optimal binary cyclic redundancy check codes with 16 parity bits crc 16 codes are presented and compared to those in existing standards for minimum distance undetected error probability on binary symmetric channels bscs and properness the codes in several cases are seen to be superior at block lengths of practical interest when they are used on low noise bscs the optimum minimum distance obtainable by some crc 16 codes is determined for all block lengths for several typical low noise bscs the minimum undetected error probability achievable with some crc 16 codes is given for all block lengths

new binary linear block codes corresp using some known techniques several new binary linear codes are constructed i e codes with a greater minimum distance than any previously known binary code of the same length and dimension 8 a detailed description of one of the constructions is given

on the undetected error probability of nonlinear binary constant weight codes the undetected error probability uep of binary n 2 spl delta m nonlinear constant weight codes over the binary symmetric channel bsc is investigated where n is the blocklength m is the weight of codeword and 2 spl delta is the minimum distance of the codes the distance distribution of the n 2 m nonlinear constant weight codes is evaluated it is proven in this paper that the 5 2 2 code 5 2 3 code 6 2 3 code 7 2 4 code 7 2 3 code and 8 2 4 code are the only proper error detecting codes in the n 2 m nonlinear constant weight codes for n spl ges 5 in the sense that their uep is increased monotonically with the channel error rate p of course all these proper codes are m out of n codes furthermore it is conjectured that except for the cases of n spl les 4 spl delta there are no proper error detecting binary n 2 spl delta m nonlinear constant weight codes for n 8 and spl deltaspl ges 1

sufficient conditions for good and proper error detecting codes the performance of linear block codes over a finite field is investigated when they are used for pure error detection sufficient conditions for a code to be good or proper for error detection are derived

poor error correction codes are poor error detection codes corresp we show that binary group codes that do not satisfy the asymptotic varshamov gilbert bound have an undesirable characteristic when used as error detection codes for transmission over the binary symmetric channel

generalized hamming weights of linear codes 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

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

evaluation of the undetected error probability of single parity check product codes in this paper an expression for the undetected error probability p epsilon of single parity check product spcp codes used for error detection over a binary symmetric channel is derived it is shown that square spcp codes need not obey a certain commonly used bound approximate expressions for the maximum p epsilon and the corresponding maximizing e are given

integral criteria for closed surfaces with constant mean curvature we propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three dimensional euclidean space these criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements we obtain from them a new proof of the old result by delaunay to the effect that the sphere is the only closed axis symmetric surface

on the probability of undetected error for linear block codes the problem of computing the probability of undetected error is considered for linear block codes used for error detection the recent literature is first reviewed and several results are extended it is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual using the dual code formulation the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes several bounds on the undetected error probability are then presented we conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained an efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results

necessary conditions for good error detection the problem of determining the error detection capabilities of linear codes where the channel is a binary symmetric channel is addressed necessary conditions are given on the number of codewords in an n k linear code and its dual for the probability of an undetected error to be upper bounded by 2 sup n k

binary linear quasi perfect codes are normal whether quasi perfect codes are normal is addressed let c be a code of length n dimension k covering radius r and minimal distance d it is proved that c is normal if d or 2r 1 hence all quasi perfect codes are normal consequently any n k r binary linear code with minimal distance d or 2r 1 is normal

on the generalized hamming weights of cyclic codes we prove several results on the generalized hamming weights ghw s of linear codes particularly for cyclic codes based on these and previously known results we give some efficient algorithms for computing ghw hierarchy of cyclic codes we give complete weight hierarchy for each of the binary cyclic codes of odd lengths spl les 31 a table of second and third ghw s of binary cyclic codes of odd lengths spl les 57 is also presented we have also computed the second ghw of all binary cyclic codes of length 63 and the third ghw of one code from each dimension

an approximation to the weight distribution of binary linear codes binary primitive bch codes form a large class of powerful error correcting codes the weight distributions of primitive bch codes are unknown except for some special classes such as the single double triple error correcting codes and some very low rate primitive bch codes however asymptotic results for the weight distribution of a large subclass of primitive bch codes have been derived by sidel nikov these results provide some insight into the weight structure of primitive bch codes sidel nikov s approach is improved and applied to the weight distribution of any binary linear block code then sidel nikov s results on the weight distributions of binary primitive bch codes are improved and it is shown that the weights of a binary primitive code have approximate binomial distribution

a method for extending binary linear codes corresp five binary linear n k d codes are presented that contain more words than the codes with corresponding values of n and d that are listed in the table of macwilliams and sloane the construction method is a generalization of construction x of sioane reddy and chen

optimization of cyclic redundancy check codes with 24 and 32 parity bits the method developed by t fujiwara et al 1985 for efficiently computing the minimum distance of shortened hamming codes using the weight distribution of their dual codes is extended to treat arbitrary shortened cyclic codes using this method implemented on a high speed special purpose processor several classes of cyclic redundancy check crc codes with 24 and 32 parity bits are investigated the crc codes of each class are known to have the same minimum distance d sub min l in a certain range l of block lengths n and within each class that crc code has been determined the minimum distance of which exceeds d sub min l up to the largest block length the d sub min profiles of the resulting codes are presented and compared with the d sub min profiles of recent suggestions of p merkey and e c posner 1984 as well as with the d sub min profile of the widely used 32 parity bit standard code recommended in ieee 802

a 55 16 19 binary goppa code and related codes having large minimum distance a 55 16 19 binary goppa code is used to construct 57 17 17 58 17 18 59 17 19 and 60 17 20 codes the first two codes have smaller redundancy than previously known codes linear or nonlinear of the same length and minimum distance the last two codes have parameters previously attained only by nonlinear codes

the weight distribution of cosets 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

on the computation of the performance probabilities for block codes with a bounded distance decoding rule when a block code is used on a discrete memoryless channel with an incomplete decoding rule that is based on a generalized distance the probability of decoding failure the probability of erroneous decoding and the expected number of symbol decoding errors can be expressed in terms of the generalized weight enumerator polynomials of the code for the symmetric erasure channel numerically stable methods to compute these probabilities or expectations are proposed for binary codes whose distance distributions are known and for linear maximum distance separable mds codes the method for linear mds codes saves the computation of the weight distribution and yields upper bounds for the probability of erroneous decoding and for the symbol error rate by the cumulative binomial distribution numerical examples include a triple error correcting bose chaudhuri hocquenghem bch code of length 63 and a reed solomon code of length 1023 and minimum distance 31

stability of hypersurfaces with constant mean curvature soit m n compacte orientable et soit x m n r n 1 une immersion a courbure moyenne constante non nulle alors x est stable si et seulement si x m r n 1 est une sphere s n r n 1

new bounds on binary linear codes of dimension eight corresp let n k d be the smallest integer n such that a binary linear code of length n dimension k and minimum distance at least d exists new results are given that improve the best previously known bounds on n 8 d

determination of the best crc codes with up to 10 bit redundancy all binary polynomials of degree up to 10 which are suitable to be used as generator polynomials of crc codes are classified and all the necessary data for the evaluation of the error control performance of the crc codes generated by the classified polynomials is calculated a procedure based on the computed data for choosing the best crc code is suggested

finding the nucleoli of large cooperative games the nucleolus is one of the most important solution concepts in cooperative game theory as a result of its attractive properties it always exists if the imputation is non empty is unique and is always in the core if the core is non empty however computing the nucleolus is very challenging because it involves the lexicographical minimization of an exponentially large number of excess values we present a method for computing the nucleoli of large games including some structured games with more than 50 players using nested linear programs lp although different variations of the nested lp formulation have been documented in the literature they have not been used for large games because of the large size and number of lps involved in addition subtle issues such as how to deal with multiple optimal solutions and with tight constraint sets need to be resolved in each lp in order to formulate and solve the subsequent ones unfortunately this technical issue has been largely overlooked in the literature we treat these issues rigorously and provide a new nested lp formulation that is smaller in terms of the number of large lps and their sizes we provide numerical tests for several games including the general flow games the coalitional skill games and the weighted voting games with up to 100 players

generalizations of the korzhik bound corresp 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

binary even 25 15 6 codes do not exist corresp that binary even 25 15 6 codes do not exist is proved by passing from coding theory to bf f 2 affine geometry all 25 sets in bf f 9 2 are shown to contain affine planes

new binary codes constructed by an old technique corresp applying a known concatenation procedure to a 30 15 12 quadratic residue code over gf 4 results in 29 new binary codes each of these codes is larger than previous codes having the same length and minimum distance

undetected error probabilities of binary primitive bch codes for both error correction and detection we investigate the undetected error probabilities for bounded distance decoding of binary primitive bch codes when they are used for both error correction and detection on a binary symmetric channel we show that the undetected error probability of binary linear codes can be simplified and quantified if the weight distribution of the code is binomial like we obtain bounds on the undetected error probability of binary primitive bch codes by applying the result to the code and show that the bounds are quantified by the deviation factor of the true weight distribution from the binomial like weight distribution

an algorithm for computing the nucleolus of disjunctive non negative additive games with an acyclic permission structure a cooperative game with a permission structure describes a situation in which players in a cooperative tu game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate in this paper we consider non negative additive games with an acyclic permission structure for such a game we provide a polynomial time algorithm for computing the nucleolus of the induced restricted game the algorithm is applied to a market situation where sellers can sell objects to buyers through a directed network of intermediaries

on the probability of undetected error for the maximum distance separable codes in this paper we investigate the performance of maximum distance separable codes with symbols from gf q when they are used for pure error detection or for simultaneous error correction and detection over a q input and q output discret memoryless channel with symbol error probability e first we show that the probability of undetected error for an mds code used for pure error detection is upper bounded by q r and decreases monotonically as edecreases from q 1 q to 0 where r is the number of parity check symbols of the code then we show that the probability of undetected error for an mds code used for correcting t or fewer symbol errors is upper bounded by q r sum min i 0 max t min i max n q 1 i and decreases monotonically as e decreases from q 1 q to 0 these results show that the mds codes are effective for both pure error detection and simultaneous error correction and detection

using codes for error correction and detection corresp a linear code c over gf q is good for t error correction and error detection if p c t epsilon leq p c t q 1 q for all epsilon 0 leq epsilon leq q 1 q where p c t epsilon is the probability of an undetected error after a codeword in c is transmitted over a q ary symmetric channel with error probability epsilon and correction is performed for all error patterns with t or 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

a new framework for designing and analyzing bist techniques and zero aliasing compression a general framework for shift register based signature analysis is presented and a mathematical model for this framework based on coding theory is developed there are two key features of this formulation first it allows for uniform treatment of lfsr misr and multiple misr based signature analyzer in addition using this formulation a new compression scheme for multiple output cut is proposed this scheme referred to as multiinput lfsr has the potential to achieve better aliasing than other schemes such as the multiple misr scheme of comparable hardware complexity several results on aliasing are presented and certain known results are shown to be direct consequences of the formulation also developed are error models that take into account the circuit topology and the effect of faults at the outputs using these models exact closed form expressions for aliasing probability are developed a closed form aliasing expression for misr under an independent error model is provided

new linear codes constructed by concatenating extending and shortening methods corresp blokh and zyablov s concatenating method constructions x y1 of sloane et al and aiitop s construction xx are applied to various codes to obtain new binary linear codes these codes improve on the recent updated helgert stinaff table of verhoeff

a 55 16 19 binary goppa code corresp a 55 16 19 binary goppa code and some related codes are described these codes are found to contain more codewords than any previously known codes of the same length and minimum distance

linear block codes for error detection corresp 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

finding and verifying the nucleolus of cooperative games the nucleolus offers a desirable payoff sharing solution in cooperative games thanks to its attractive properties it always exists and lies in the core if the core is non empty and it is unique the nucleolus is considered as the most stable solution in the sense that it lexicographically minimizes the dissatisfactions among all coalitions although computing the nucleolus is very challenging the kohlberg criterion offers a powerful method for verifying whether a solution is the nucleolus in relatively small games i e with the number of players n leq 15 this approach however becomes more challenging for larger games because of the need to form and check a criterion involving possibly exponentially large collections of coalitions with each collection potentially of an exponentially large size the aim of this work is twofold first we develop an improved version of the kohlberg criterion that involves checking the balancedness of at most n 1 sets of coalitions second we exploit these results and introduce a novel descent based constructive algorithm to find the nucleolus efficiently we demonstrate the performance of the new algorithms by comparing them with existing methods over different types of games our contribution also includes the first open source code for computing the nucleolus for games of moderately large sizes

the concatenated structure of cyclic and abelian codes it is shown that a linear two dimensional cyclic code of length nn can be factorized into a direct sum of concatenated codes with cyclic inner and outer codes and conversely thai a two dimensional cyclic code can be constructed in this way this result is extended and it appears that the abelian codes are obtainable by taking a direct sum of several concatenations of cyclic codes codes are constructed that are better than any previously known in particular low rate cyclic codes superior to the duals of high rate bch codes are constructed

on testing for improper error detection codes a family of tests for improper codes is given these tests can be used in cases where the complete weight distribution of the code is unknown it was found that knowledge of the number of minimum weight codewords can be used to greatly increase the effectiveness of the asymptotic varshamov gilbert test further improvement is possible as more is known about the number of other weight codewords

minimum support weights of binary codes some relations between the minimum support weights are discussed in particular the possible weight hierarchies of codes of dimension 4 are determined

a note upon another error detecting code that is not proper in this correspondence the performance of the nonlinear error detecting code described in ccir recommendation 476 2 when used over a memoryless binary symmetric channel bsc is derived from this it is apparent that the probability of accepting an erroneous block is not a monotonic function of the bit error probability but rather reaches a peak of 5 653 percent when the bit error probability is 11 66 percent consequently this code is not proper 1 6

on the probability of undetected error for overextended reed solomon codes upper and lower bounds on the weight distribution of overextended reed solomon oers codes are derived from which tight upper and lower bounds on the probability of undetected error for oers codes are obtained for q ary symmetric channels