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

a simple derivation of the undetected error probabilities of complementary codes recently fu klove and wei 2003 have shown that the undetected error probability of a binary code is related to that of its complement and the undetected error probability of a constant weight binary code is related to that of its complement relative to the set of all constant weight vectors we generalize these relations to cover the complements of any binary or nonbinary code relative to a distance invariant code containing the first code we prove the generalization using a much simpler argument than the published proofs of the special cases

a lower bound on the probability of undetected error for binary constant weight codes in this correspondence we study the probability of undetected error for binary constant weight codes first we derive a new lower bound on the probability of undetected error for binary constant weight codes next we show that this bound is tight if and only if the binary constant weight codes are generated from certain t designs in combinatorial design theory this means that these binary constant weight codes generated from certain t designs are uniformly optimal for error detection along the way we determine the distance distributions of such binary constant weight codes in particular it is shown that binary constant weight codes generated from steiner systems are uniformly optimal for error detection thus we prove a conjecture of xia fu jiang and ling furthermore the distance distribution of a binary constant weight code generated from a steiner system is determined finally we study the exponent of the probability of undetected error for binary constant weight codes we derive some bounds on the exponent of the probability of undetected error for binary constant weight codes these bounds enable us to extend the region in which the exponent of the probability of undetected error is exactly determined

association schemes and coding theory this paper contains a survey of association scheme theory with its algebraic and analytical aspects and of its applications to coding theory in a wide sense it is mainly concerned with a class of subjects that involve the central notion of the distance distribution of a code special emphasis is put on the linear programming method inspired by the macwilliams transform this produces upper bounds for the size of a code with a given minimum distance and lower bounds for the size of a design with a given strength the most specific results are obtained in the case where the underlying association scheme satisfies certain well defined polynomial properties this leads one into the realm of orthogonal polynomial theory in particular some universal bounds are derived for codes and designs in polynomial type association schemes throughout the paper the main concepts methods and results are illustrated by two examples that are of major significance in classical coding theory namely the hamming scheme and the johnson scheme other topics that receive special attention are spherical codes and designs and additive codes in translation schemes including z sub 4 additive binary codes

optical orthogonal codes design analysis and applications an optical orthogonal code ooc is a family of 0 1 sequences with good auto and cross correlation properties i e the autocorrelation of each sequence exhibits the thumbtack shape and the cross correlation between any two sequences remains low throughout the use of optical orthogonal codes enables a large number of asynchronous users to transmit information efficiently and reliably the thumbtack shaped autocorrelation facilitates the detection of the desired signal and low profile cross correlation reduces interference from unwanted signals theoretical upper and lower bounds on the maximum possible size of oocs are derived methods for the design and analysis of oocs using tools from projective geometry the greedy algorithm iterative constructions algebraic coding theory block design and various other combinational disciplines are discussed

semi localizations of semi abelian categories abstract a semi localization of a category is a full reflective subcategory with the property that the reflector is semi left exact there are many interesting examples of semi localizations as for instance any torsion free subcategory of a semi abelian category by specializing a result due to s mantovani we first characterize the categories which are semi localizations of exact mal tsev categories we then prove a new characterization of protomodular categories in terms of binary relations allowing us to obtain an abstract characterization of the semi localizations of exact protomodular categories this result is very useful to study the hereditarily torsion free subcategories of semi abelian categories some examples are considered in detail in the categories of groups crossed modules commutative rings and topological groups we finally explain how these results extend the corresponding ones obtained in the abelian context by w rump

the undetected error probability threshold of m out of n codes 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

torsion theories in homological categories we introduce and study the notion of torsion theory in the non abelian context of homological categories and we investigate the properties of the corresponding closure operator we then consider several new examples of torsion theories in the category of topological groups and more generally in any category of topological semi abelian algebras we finally characterize the hereditary torsion theories and we analyse a new example in the homological category of crossed modules c 2006 elsevier inc all rights reserved

torsion theories and radicals in normal categories we introduce a relativized notion of semi normalcy for categories that come equipped with a proper stable factorization system and we use radicals in the sense of module theory and normal closure operators in order to study torsion theories in such categories our results generalize and complement recent studies in the realm of semi abelian and in part homological categories in particular we characterize both torsion free and torsion classes in terms of their closure under extensions we pay particular attention to the homological and for our purposes more importantly normal categories of topological algebra such as the category of topological groups but our applications go far beyond the realm of these types of categories as they include for example the normal but non homological category of pointed topological spaces which is in fact a rich supplier for radicals of topological groups

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

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

recursion formulas for the lie integral abstract an explicit definition is given of a non linear integral which is termed the lie integral this integral is defined for lie algebra valued functions in terms of the representations of the algebra various properties of the integral are investigated and applications are given to differential equations applications to differential geometry will be given elsewhere

topological semi abelian algebras given an algebraic theory t whose category of models is semi abelian we study the category top t of topological models of t and generalize to it most classical results on topological groups in particular top t is homological which includes barr regularity and forces the mal cev property every open subalgebra is closed and every quotient map is open we devote special attention to the hausdorff compact locally compact connected totally disconnected and profinite t algebras c 2004 elsevier inc all rights reserved

the probability of undetected error for binary constant weight codes in this correspondence we study the probability of undetected error for binary constant weight codes first we derive a new formula on the probability of undetected error for binary constant weight codes second using this new formula and linear programming we give two new lower bounds on the probability of undetected error for binary constant weight codes these two new lower bounds improve on previously known lower bounds in certain cases furthermore we show that these two lower bounds are tight if and only if the binary constant weight codes are generated from certain t designs in combinatorial design theory this means that these binary constant weight codes generated from certain t designs are uniformly optimal for error detection along the way we determine the distance distributions of such binary constant weight codes finally several examples are given to illustrate the results obtained in this correspondence

a poincare lemma for connection forms denote by p the space of piecewise smooth curves in rn beginning at the origin a path 2 form is a function h on p such that for each element in p h is a 2 form at the endpoint of with values in a lie algebra g for example if a is a smooth g valued connection form on rn with curvature f and parallel translation operator p then the equation la p 1 f 1 p defines la as a path 2 form a necessary and sufficient condition is given to characterize those path 2 forms which arise in this way by way of application it is shown that the birula mandelstam generalization of maxwell s equations to nonabelian gauge fields is equivalent to the yang mills equation

binary constant weight codes for error detection the undetected error probability of binary constant weight codes on a binary symmetric channel is studied in this correspondence first we present a necessary condition for the binary constant weight codes to be proper for error detection then by using the necessary condition we study the error detection capability of binary optimum constant weight codes

fundamental group functors in descent exact homological categories abstract we study the notion of a fundamental group in the framework of descent exact homological categories this setting is sufficiently large to include several categories of algebraic nature such as almost abelian categories semi abelian categories and categories of topological semi abelian algebras for many adjunctions in this context we describe the fundamental groups by generalised brown ellis hopf formulae for the integral homology of groups