spherically punctured biorthogonal codes consider a binary reed muller code rm r m defined on the hypercube bb f2m and let all code positions be restricted to the m tuples of a given hamming weight b in this paper we specify this single layer construction obtained from the biorthogonal codes rm 1 m and the hadamard codes h m both punctured codes inherit some recursive properties of the original rm codes however they cannot be formed by the recursive plotkin construction we first observe that any code vector in these codes has hamming weight defined by the weight w of its information block more specifically this weight depends on the absolute values of the krawtchouk polynomials kbm w we then study the properties of the krawtchouk polynomials and show that the minimum code weight of a single layer code rm 1 m b is achieved at the minimum input weight w 1 for any we further refine our codes by limiting the possible weights w of the input information blocks as a result some of the designed code sequences meet or closely approach the griesmer bound finally we consider more general punctured codes whose positions form several spherical layers

propagation of singularities for semilinear hyperbolic initial boundary value problems in one space dimension on etudie la propagation et la reflexion de la regularite pour des systemes semi lineaires strictement hyperboliques a 1 variable d espace

analysis of the block error probability of concatenated polar code ensembles in this paper we provide an analysis of the performance of concatenation of polar codes with outer cyclic redundancy check crc codes separated by an interleaver in the short and moderate block length regimes the analysis addresses maximum likelihood decoding as a proxy to the code performance under successive cancellation list decoding the analysis is carried out by introducing the concatenated polar code cpc ensembles whose distance properties can be analyzed for sufficiently short block lengths by means of the uniform interleaver approach at moderate block lengths we resort to the monte carlo simulations results show that if the inner polar code possesses a low minimum distance and the outer crc code has a sufficiently large amount of redundancy then the choice of the outer code generator polynomial and the interleaver may yield to a large variability in the performance of the resulting cpc

krawtchouk polynomials and universal bounds for codes and designs in hamming spaces universal bounds for the cardinality of codes in the hamming space f sub r sup n with a given minimum distance d and or dual distance d are stated a self contained proof of optimality of these bounds in the framework of the linear programming method is given the necessary and sufficient conditions for attainability of the bounds are found the parameters of codes satisfying these conditions are presented in a table a new upper bound for the minimum distance of self dual codes and a new lower bound for the crosscorrelation of half linear codes are obtained

interleaved polar i polar codes by inserting interleavers between intermediate stages of the polar encoder a new class of polar codes termed interleaved polar i polar codes is proposed by the uniform interleaver assumption we derive the weight enumerating function wef and the input output weight enumerating function iowef averaged over the ensemble of i polar codes the average wef can be used to calculate the upper bound on the average block error rate bler of a code selected at random from the ensemble of i polar codes also we propose a concatenated coding scheme that employs p high rate codes as the outer code and q i polar codes as the inner code with an interleaver in between the average wef of the concatenated code is derived based on the uniform interleaver assumption simulation results show that bler upper bounds can well predict bler performance levels of the concatenated codes the results show that the performance of the proposed concatenated code with p q 2 is better than that of the crc aided i polar code with p q 1 of the same length and code rate at high signal to noise ratios snrs moreover the proposed concatenated code allows multiple decoders to operate in parallel which can reduce the decoding latency and hence is suitable for ultra reliable low latency communications urllc

list decoding of polar codes we describe a successive cancellation list decoder for polar codes which is a generalization of the classic successive cancellation decoder of arikan in the proposed list decoder l decoding paths are considered concurrently at each decoding stage where l is an integer parameter at the end of the decoding process the most likely among the l paths is selected as the single codeword at the decoder output simulations show that the resulting performance is very close to that of maximum likelihood decoding even for moderate values of l alternatively if a genie is allowed to pick the transmitted codeword from the list the results are comparable with the performance of current state of the art ldpc codes we show that such a genie can be easily implemented using simple crc precoding the specific list decoding algorithm that achieves this performance doubles the number of decoding paths for each information bit and then uses a pruning procedure to discard all but the l most likely paths however straightforward implementation of this algorithm requires omega l n 2 time which is in stark contrast with the o n log n complexity of the original successive cancellation decoder in this paper we utilize the structure of polar codes along with certain algorithmic transformations in order to overcome this problem we devise an efficient numerically stable implementation of the proposed list decoder that takes only o l n log n time and o l n space

on a coupled nonlinear singular thermoelastic system abstract for a coupled nonlinear singular system of thermoelasticity with one space dimension we consider its initial boundary value problem on an interval for one of the unknowns a classical condition is replaced by a nonlocal constraint of integral type because of the presence of a memory term in one of the equations and the presence of a weighted boundary integral condition the solution requires a delicate set of techniques we first solve a particular case of the given nonlinear problem by using a functional analysis approach on the basis of the results obtained and an iteration method we establish the well posedness of solutions in weighted sobolev spaces

spherically punctured reed muller codes consider a binary reed muller code rm r m defined on the m dimensional hypercube mathbb f 2 m in this paper we study punctured reed muller codes p r m b whose positions are restricted to the m tuples of a given hamming weight b in combinatorial terms this paper concerns m variate boolean polynomials of any degree r which are evaluated on a hamming sphere of some radius b in mathbb f 2 m codes p r m b inherit some recursive properties of rm codes in particular they can be built from the shorter codes by decomposing a spherical b layer into sub layers of smaller dimensions however these sub layers have different sizes and do not form the classical plotkin construction we analyze recursive properties of the spherically punctured codes p r m b and find their distances for the arbitrary values of parameters r m and b finally we describe recursive successive cancellation decoding of these codes

weight distribution and list decoding size of reed muller codes the weight distribution and list decoding size of reed muller codes are studied in this work given a weight parameter we are interested in bounding the number of reed muller codewords with weight up to the given parameter and given a received word and a distance parameter we are interested in bounding the size of the list of reed muller codewords that are within that distance from the received word obtaining tight bounds for the weight distribution of reed muller codes has been a long standing open problem in coding theory dating back to 1976 in this work we make a new connection between computer science techniques used to study low degree polynomials and these coding theory questions this allows us to resolve the weight distribution and list decoding size of reed muller codes for all distances previous results could only handle bounded distances azumi kasami and tokura gave bounds on the weight distribution which hold up to 2 5 times the minimal distance of the code and gopalan klivans and zuckerman gave bounds on the list decoding size which hold up to the johnson bound

relaxed polar codes polar codes are the latest breakthrough in coding theory as they are the first family of codes with explicit construction that provably achieve the symmetric capacity of binary input discrete memoryless channels polar encoding and successive cancellation decoding have the complexities of n log n for code length n although the complexity bound of n log n is asymptotically favorable we report in this work methods to further reduce the encoding and decoding complexities of polar coding the crux is to relax the polarization of certain bit channels without performance degradation we consider schemes for relaxing the polarization of both very good and very bad bit channels in the process of channel polarization relaxed polar codes are proved to preserve the capacity achieving property of polar codes analytical bounds on the asymptotic and finite length complexity reduction attainable by relaxed polarization are derived for binary erasure channels we show that the computation complexity can be reduced by a factor of six while preserving the rate and error performance we also show that relaxed polar codes can be decoded with significantly reduced latency for additive white gaussian noise channels with medium code lengths we show that relaxed polar codes can have lower error probabilities than conventional polar codes while having reduced encoding and decoding computation complexities

randomized polar subcodes with optimized error coefficient a method for construction of polar subcodes with reduced error coefficient is presented the proposed approach relies on explicit enumeration of low weight non zero codewords in a polar code and construction of dynamic freezing constraints which define a subcode not containing most of these codewords the obtained codes provide a large performance gain in the high snr region compared to non optimized polar subcodes and polar codes with crc

ratio list decoding we extend the notion of list decoding to ratio list decoding which involves a list decoder whose list size is specified as a function of the number of messages m n and the block length n we present the necessary and sufficient conditions on m n for the existence of code sequences which enable reliable list decoding with respect to the desired list size l m n n it is shown that the ratio capacity defined as the supremum of achievable normalized logarithms of the ratio r m n n m n l m n n is equal to the shannon channel capacity c for both stochastic and deterministic encoding allowing for random list size we are able to deduce some properties of identification codes where the decoder s output can be viewed as a list of messages corresponding to decision regions that include the channel output we further address the regime of mismatched list decoding in which the list constitutes of the codewords that accumulate the highest score values jointly with the channel output according to some given function we study the case of deterministic encoding and mismatched ratio list decoding we establish similar necessary and sufficient conditions for the existence of code sequences which enable reliable mismatched list decoding with respect to the desired list size l m n n and we show that the ratio capacity with mismatched decoding is equal to the mismatch capacity focusing on the case of an exponential list size l n e n theta its comparison with ordinary mismatched decoding shows that the increase in capacity is by theta bits per channel used for all channels and decoding metrics several properties of the average error probability in the setup of mismatched list decoding with deterministic list size are provided

further classifications of codes meeting the griesmer bound corresp for any n k d binary linear code the griesmer bound says that n geq sum i 0 k 1 lceil d 2 i rceil where lceil x rceil denotes the smallest integer geq x we consider codes meeting the griesmer bound with equality these codes have parameters left s 2 k 1 sum i 1 p 2 u i 1 k s2 k 1 sum i 1 p 2 u i 1 right where k u 1 cdots u p geq 1 we characterize all such codes when p 2 or u i 1 u i geq 2 for 2 leq i leq p

propagation of singularities in one dimensional thermoelasticity the propagation of singularities for the system of homogeneous thermoelasticity in one space dimension is studied linear and a class of semilinear cauchy problems are considered

global smooth solutions and asymptotic stability in one dimensional nonlinear thermoelasticity the global existence of smooth solutions to the equations of nonlinear thermoelasticity is shown for a one dimensional homogeneous reference configuration dirichlet boundary conditions are studied and the asymptotic behaviour of the solutions as t is described

pseudorandom bits for polynomials we present a new approach to constructing pseudorandom generators that fool low degree polynomials over finite fields based on the gowers norm using this approach we obtain the following main constructions of explicitly computable generators g mathbb f s to mathbb f n that fool polynomials over a finite field mathbb f we stress that the results in 1 and 2 are unconditional i e do not rely on any unproven assumption moreover the results in 3 rely on a special case of the conjecture which may be easier to prove our generator for degree d polynomials is the componentwise sum of d generators for degree 1 polynomials on independent seeds prior to our work generators with logarithmic seed length were only known for degree 1 i e linear polynomials j naor and m naor siam j comput 22 1993 pp 838 856 in fact over small fields such as mathbb f 2 0 1 our results constitute the first progress on these problems since the long standing generator by luby velickovic and wigderson deterministic approximate counting of depth 2 circuits in proceedings of the 2nd israeli symposium on theoretical computer science istcs 1993 pp 18 24 whose seed length is much bigger s exp left omega left sqrt log n right right even for the case of degree 2 polynomials over mathbb f 2

channel coding rate in the finite blocklength regime this paper investigates the maximal channel coding rate achievable at a given blocklength and error probability for general classes of channels new achievability and converse bounds are given which are tighter than existing bounds for wide ranges of parameters of interest and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100 it is also shown analytically that the maximal rate achievable with error probability isclosely approximated by c v n q 1 where c is the capacity v is a characteristic of the channel referred to as channel dispersion and q is the complementary gaussian cumulative distribution function

on the finite length scaling of q ary polar codes the polarization process of polar codes over a prime q ary alphabet is studied recently it has been shown that the blocklength of polar codes with prime alphabet size scales polynomially with respect to the inverse of the gap between code rate and channel capacity however except for the binary case the degree of the polynomial in the bound is extremely large in this paper a different approach to computing the degree of this polynomial for any prime alphabet size is shown this approach yields a lower degree polynomial for various values of the alphabet size that were examined it is also shown that even lower degree polynomial can be computed with an additional numerical effort

weight distributions for successive cancellation decoding of polar codes in this paper we derive the exact weight distributions that emerge during each stage of successive cancellation decoding of polar codes though we do not compute the distance spectrum of polar codes the results allow us to get an estimate of the decoding error probability and to show a link between the first nonzero components of the weight distribution and the partial order between the synthetic channels also we establish the minimal distance between two cosets associated with two paths that differ in two positions this can be regarded as a first step toward analyzing the weight distributions for successive cancellation list decoding

on distance properties of convolutional polar codes a lower bound on the minimum distance of convolutional polar codes is provided the bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix moreover a construction of convolutional polar subcodes is proposed which provides improved performance under successive cancellation list decoding for sufficiently large list size the decoding complexity of convolutional polar subcodes appears to be lower compared with arikan polar subcodes with the same performance the error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of arikan polar subcodes with the same list size

soft decision decoding of reed muller codes recursive lists recursive list decoding is considered for reed muller rm codes the algorithm repeatedly relegates itself to the shorter rm codes by recalculating the posterior probabilities of their symbols intermediate decodings are only performed when these recalculations reach the trivial rm codes in turn the updated lists of most plausible codewords are used in subsequent decodings the algorithm is further improved by using permutation techniques on code positions and by eliminating the most error prone information bits simulation results show that for all rm codes of length 256 and many subcodes of length 512 these algorithms approach maximum likelihood ml performance within a margin of 0 1 db as a result we present tight experimental bounds on ml performance for these codes

error exponents for two soft decision decoding algorithms of reed muller codes error exponents are studied for recursive and majority decoding of general reed muller rm codes rm r m used on the additive white gaussian noise awgn channels both algorithms have low complexity and correct many error patterns whose weight exceeds half the code distance decoding consists of multiple consecutive steps which repeatedly recalculate the input symbols and determine different information symbols using soft decision majority voting for any code rm r m we estimate the probabilities of the information symbols obtained in these recalculations and derive the analytical upper bounds for the block error rates of the recursive and majority decoding in the case of a low noise we also obtain the lower bounds and show that the upper bounds are tight for a higher noise these bounds closely approach our simulation results

propagation of singularities in cauchy problems for quasilinear thermoelastic systems in three space variables by using microlocal analysis the propagation of weak singularities in cauchy problems for quasilinear thermoelastic systems in three space variables are investigated first paradifferential operators are employed to decouple the quasilinear thermoelastic systems second by investigating the decoupled hyperbolic parabolic systems and using the classical bootstrap argument the property of finite propagation speeds of singularities in cauchy problems for the quasilinear thermoelastic systems is obtained finally it is shown that the microlocal weak singularities for cauchy problems of the thermoelastic systems are propagated along the null bicharacteristics of the hyperbolic operators

how to construct polar codes a method for efficiently constructing polar codes is presented and analyzed although polar codes are explicitly defined straightforward construction is intractable since the resulting polar bit channels have an output alphabet that grows exponentially with the code length thus the core problem that needs to be solved is that of faithfully approximating a bit channel with an intractably large alphabet by another channel having a manageable alphabet size we devise two approximation methods which sandwich the original bit channel between a degraded and an upgraded version thereof both approximations can be efficiently computed and turn out to be extremely close in practice we also provide theoretical analysis of our construction algorithms proving that for any fixed e 0 and all sufficiently large code lengths n polar codes whose rate is within e of channel capacity can be constructed in time and space that are both linear in n

unified scaling of polar codes error exponent scaling exponent moderate deviations and error floors consider the transmission of a polar code of block length n and rate r over a binary memoryless symmetric channel w and let p mathrm e be the block error probability under successive cancellation decoding in this paper we develop new bounds that characterize the relationship of the parameters r n p mathrm e and the quality of the channel w quantified by its capacity i w and its bhattacharyya parameter z w in previous work two main regimes were studied in the error exponent regime the channel w and the rate r are fixed and it was proved that the error probability p mathrm e scales roughly as 2 sqrt n in the scaling exponent approach the channel w and the error probability p mathrm e are fixed and it was proved that the gap to capacity i w r scales as n 1 mu here mu is called scaling exponent and this scaling exponent depends on the channel w a heuristic computation for the binary erasure channel bec gives mu 3 627 and it was shown that for any channel w 3 579 le mu le 5 702 our contributions are as follows first we provide the tighter upper bound mu le 4 714 valid for any w with the same technique we obtain the upper bound mu le 3 639 for the case of the bec this upper bound approaches very closely the heuristically derived value for the scaling exponent of the erasure channel second we develop a trade off between the gap to capacity i w r and the error probability p mathrm e as the functions of the block length n in other words we neither fix the gap to capacity error exponent regime nor the error probability scaling exponent regime but we do consider a moderate deviations regime in which we study how fast both quantities as the functions of the block length n simultaneously go to 0 third we prove that polar codes are not affected by error floors to do so we fix a polar code of block length n and rate r then we vary the channel w and study the impact of this variation on the error probability we show that the error probability p mathrm e scales as the bhattacharyya parameter z w raised to a power that scales roughly like sqrt n this agrees with the scaling in the error exponent regime

soft decision decoding of reed muller codes a simplified algorithm soft decision decoding is considered for general reed muller rm codes of length n and distance d used over a memoryless channel a recursive decoding algorithm is designed and its decoding threshold is derived for long rm codes the algorithm has complexity of order nlnn and corrects most error patterns of the euclidean weight of order radicn lnn instead of the decoding threshold radicd 2 of the bounded distance decoding also for long rm codes of fixed rate r the new algorithm increases 4 pi times the decoding threshold of its hard decision counterpart

dynamic scflip decoding of polar codes this paper proposes a generalization of the recently introduced successive cancellation flip scflip decoding of polar codes characterized by a number of extra decoding attempts where one or several positions are flipped from the standard sc decoding to make such an approach effective we first introduce the concept of higher order bit flips and propose a new metric to determine the bit flips that are more likely to correct the trajectory of the sc decoding we then propose a generalized scflip decoding algorithm referred to as dynamic scflip d scflip which dynamically builds a list of candidate bit flips while guaranteeing that the next attempt has the highest probability of success among the remaining ones simulation results show that d scflip is an effective alternative to sc list decoding of polar codes by providing very good error correcting performance with an average computation complexity close to the one of the sc decoder

soft decision majority decoding of reed muller codes we present a new soft decision majority decoding algorithm for reed muller codes rm r m first the reliabilities of 2 sup m transmitted symbols are recalculated into the reliabilities of 2 sup m r parity checks that represent each information bit in turn information bits are obtained by the weighted majority that gives more weight to more reliable parity checks it is proven that for long low rate codes rm r m our soft decision algorithm outperforms its conventional hard decision counterpart by 10 log sub 10 spl pi 2 spl ap 2 db at any given output error probability for fixed code rate r and m spl rarr spl infin our algorithm increases almost 2 sup r 2 times the correcting capability of soft decision bounded distance decoding

learning decision trees using the fourier spectrum this work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution this algorithm uses membership queries the decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node i e summation of a subset of the input variables over gf 2 this paper shows how to learn in polynomial time any function that can be approximated in norm l 2 by a polynomially sparse function i e a function with only polynomially many nonzero fourier coefficients the authors demonstrate that any function f whose l 1 norm i e the sum of absolute value of the fourier coefficients is polynomial can be approximated by a polynomially sparse function and prove that boolean decision trees with linear operations are a subset of this class of functions moreover it is shown that the functions with polynomial l 1 norm can be learned deterministically the algorithm can also exactly identi

the rank of a finite primitive solvable permutation group let p be a prime and n a positive integer and let gf p denote the finite field with p elements the group s p x ax b i a b e gf p a 0 0 e aut gf p 1s a solvable group of derived length 3 which acts 2 transitively on gf p and the stabilizer of 0 in s p is t p x ax 1 a e gf p a 0 cr e aut gf p conversely huppert 4 in 1957 showed that with the exception of certain groups of degree 32 52 72 1 12 232 and 34 every finite solvable 2 transitive group is permutation isomorphic to a subgroup of some s p with this action the proof involves the study of primitive solvable permutation groups let g be a finite primitive solvable permutation group of degree d then see e g 4 the minimal normal subgroup v of g is regular and elemen tary abelian of order d p for some prime p and a one point stabilizer g of g acts on v by conjugation as an irreducible subgroup of the linear group gl n p by definition the rank r of g is the number of orbits of the stabilizer g since the 2 transitive groups are precisely the primitive groups of rank 2 huppert could prove his result by classifying the irreducible linear groups with 2 orbits in 1969 foulser 3 generalized huppert s result to primitive solvable permutation groups of rank 3 or 4 for this it was necessary to consider separately the cases where the stabilizer g is primitive and where it is imprimitive as a linear group in the case that the stabilizer g is a primitive linear group dornhoff 2 proved the existence of a function f r such that if g has rank r then either g is permutation isomorphic to a subgroup of some s p or g has degree at most f r without attempting to find any realistic estimate for the values of j we quantify dornhoffs result as follows

reed muller codes achieve capacity on erasure channels we introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding rather than relying on the precise structure of the codes our method exploits code symmetry in particular the technique applies to any sequence of linear codes where the blocklengths are strictly increasing the code rates converge and the permutation group of each code is doubly transitive in other words we show that symmetry alone implies near optimal performance an important consequence of this result is that a sequence of reed muller codes with increasing blocklength and converging rate achieves capacity this possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1 moreover these results extend naturally to all affine invariant codes and thus to extended primitive narrow sense bch codes this also resolves in the affirmative the existence question for capacity achieving sequences of binary cyclic codes the primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions

performance bounds of concatenated polar coding schemes a concatenated coding scheme over binary memoryless symmetric bms channels using a polarization transformation followed by outer sub codes is analyzed achievable error exponents and upper bounds on the error rate are derived the first bound is obtained using outer codes which are typical random linear codes as a byproduct of this bound it determines the rates of the outer codes a lower bound on the error exponent that holds for all bms channels with a given capacity is then derived improved bounds and approximations for finite blocklength codes using channel dispersions normal approximation as well as converse and approximate converse results are also obtained the bounds are compared with actual simulation results from the literature for the cases considered when transmitting over the binary input additive white gaussian noise channel there was only a small gap between the channel dispersion based approximation and the actual error rate of concatenated bch polar codes

remarks on propagation of singularities in thermoelasticity the propagation of high order weak singularities for the system of homogeneous thermoelasticity in one space variable is studied by using paralinearization and a new decoupling technique introduced by the author microlocal analysis in nonlinear thermoelasticity to appear for the linear system one shows that the nonsmooth initial data for the parabolic part lead to singularities in the hyperbolic part of solutions even when the initial data for that part are identically zero both the cauchy problem and the problem inside of a domain for the semilinear system are considered it is shown that the propagation of high order singularities is essentially dominated by the hyperbolic operator in the system of thermoelasticity

soft decision decoding of reed muller codes as generalized multiple concatenated codes constructs reed muller codes by generalized multiple concatenation of binary block codes of length 2 as a consequence of this construction a new decoding procedure is derived that uses soft decision information the algorithm is designed for low decoding complexity and is applicable to all reed muller codes it gives better decoding performance than soft decision bounded distance decoding its decoding complexity is much lower than that of maximum likelihood trellis decoding of reed muller codes especially for long codes

construction of polar codes with sublinear complexity consider the problem of constructing a polar code of block length n for a given transmission channel w previous approaches require one to compute the reliability of the n synthetic channels and then use only those that are sufficiently reliable however we know from two independent works by schurch and by bardet et al that the synthetic channels are partially ordered with respect to degradation hence it is natural to ask whether the partial order can be exploited to reduce the computational burden of the construction problem we show that if we take advantage of the partial order we can construct a polar code by computing the reliability of roughly a fraction 1 log 3 2 n of the synthetic channels in particular we prove that n log 3 2 n is a lower bound on the number of synthetic channels to be considered and such a bound is tight up to a multiplicative factor log log n this set of roughly n log 3 2 n synthetic channels is universal in the sense that it allows one to construct polar codes for any w and it can be identified by solving a maximum matching problem on a bipartite graph our proof technique consists of reducing the construction problem to the problem of computing the maximum cardinality of an antichain for a suitable partially ordered set as such this method is general and it can be used to further improve the complexity of the construction problem in case a refined partial order on the synthetic channels of polar codes is discovered

propagation of mild singularities in higher dimensional thermoelasticity the propagation of mild singularities for the semilinear model of three dimensional thermoelasticity is studied it is shown that the propagation picture of such singularities of the solution to the semilinear model coincides with one of the solutions to the corresponding linear model as a simple consequence of our method a similar result for the full semilinear cauchy problem of one dimensional thermoelasticity is also presented

an improved sphere packing bound for finite length codes over symmetric memoryless channels this paper derives an improved sphere packing isp bound for finite length error correcting codes whose transmission takes place over symmetric memoryless channels and the codes are decoded with an arbitrary list decoder we first review classical results i e the 1959 sphere packing sp59 bound of shannon for the gaussian channel and the 1967 sphere packing sp67 bound of shannon et al for discrete memoryless channels an improvement on the sp67 bound as suggested by valembois and fossorier is also discussed these concepts are used for the derivation of a new lower bound on the error probability of list decoding referred to as the isp bound which is uniformly tighter than the sp67 bound and its improved version the isp bound is applicable to symmetric memoryless channels and some of its applications are presented its tightness under maximum likelihood ml decoding is studied by comparing the isp bound to previously reported upper and lower bounds on the ml decoding error probability and also to computer simulations of iteratively decoded turbo like codes this paper also presents a technique which performs the entire calculation of the sp59 bound in the logarithmic domain thus facilitating the exact calculation of this bound for moderate to large block lengths without the need for the asymptotic approximations provided by shannon

simple map decoding of first order reed muller and hamming codes a maximum a posteriori map probability decoder of a block code minimizes the probability of error for each transmitted symbol separately the standard way of implementing map decoding of a linear code is the bahl cocke jelinek raviv bcjr algorithm which is based on a trellis representation of the code the complexity of the bcjr algorithm for the first order reed muller rm 1 codes and hamming codes is proportional to n sup 2 where n is the code s length in this correspondence we present new map decoding algorithms for binary and nonbinary rm 1 and hamming codes the proposed algorithms have complexities proportional to q sup 2 n log sub q n where q is the alphabet size in particular for the binary codes this yields complexity of order n log n

polar codes characterization of exponent bounds and constructions polar codes were recently introduced by arikan they achieve the symmetric capacity of arbitrary binary input discrete memoryless channels under a low complexity successive cancellation decoding scheme the original polar code construction is closely related to the recursive construction of reed muller codes and is based on the 2 2 matrix 1 0 1 1 it was shown by arikan telatar that this construction achieves an error exponent of 1 2 i e that for sufficiently large blocklengths the error probability decays exponentially in the square root of the blocklength it was already mentioned by arikan that in principle larger matrices can be used to construct polar codes in this paper it is first shown that any l l matrix none of whose column permutations is upper triangular polarizes binary input memoryless channels the exponent of a given square matrix is characterized upper and lower bounds on achievable exponents are given using these bounds it is shown that there are no matrices of size smaller than 15 15 with exponents exceeding 1 2 further a general construction based on bch codes which for large i achieves exponents arbitrarily close to 1 is given at size 16 16 this construction yields an exponent greater than 1 2

reed muller codes for random erasures and errors this paper studies the parameters for which binary reed muller rm codes can be decoded successfully on the binary erasure channel and binary symmetry channel and in particular when can they achieve capacity for these two classical channels necessarily this paper also studies the properties of evaluations of multivariate gf 2 polynomials on the random sets of inputs for erasures we prove that rm codes achieve capacity both for very high rate and very low rate regimes for errors we prove that rm codes achieve capacity for very low rate regimes and for very high rates we show that they can uniquely decode at about the square root of the number of errors at capacity the proofs of these four results are based on different techniques which we find interesting in their own right in particular we study the following questions about e m r the matrix whose rows are the truth tables of all the monomials of degree leq r in m variables what is the most resp least number of random columns in e m r that define a submatrix having full column rank resp full row rank with high probability we obtain tight bounds for very small resp very large degrees r which we use to show that rm codes achieve capacity for erasures in these regimes our decoding from random errors follows from the following novel reduction for every linear code c of sufficiently high rate we construct a new code c obtained by tensorizing c such that for every subset s of coordinates if c can recover from erasures in s then c can recover from errors in s specializing this to the rm codes and using our results for erasures imply our result on the unique decoding of the rm codes at high rate finally two of our capacity achieving results require tight bounds on the weight distribution of rm codes we obtain such bounds extending the recent bounds from constant degree to linear degree polynomials

on the number of conjugacy classes of a finite solvable group ii abstract we prove that a finite solvable group g has at least 49p 1 60 conjugacy classes whenever p is a prime such that p2 divides the order of g we also construct an infinite family of finite solvable groups where this bound is attained

a lower bound for the number of conjugacy classes of a finite group abstract every finite group whose order is divisible by a prime p has at least 2 p 1 conjugacy classes

strong asymptotics for krawtchouk polynomials we determine the strong asymptotics for the class of krawtchouk polynomials on the real line we show how our strong asymptotics describes the limiting distribution of the zeros of the krawtchouk polynomials

microlocal analysis in nonlinear thermoelasticity abstract this paper is devoted to the study of the propagation of singularities for semilinear hyperbolic parabolic coupled systems in thermoelasticity first a linear transformation involving pseudodifferential operators for unknowns is introduced to decouple the hyperbolic and the parabolic operators by using the decoupled system we obtain that the semilinear system of thermoelasticity has finite speeds for the propagation of singularities furthermore it is proved that the microlocal singularities of solutions to this semilinear hyperbolic parabolic coupled system are propagated along null bicharacteristics of the hyperbolic operator by using the theory of nonsmooth pseudodifferential operators finally the cauchy problem for the semilinear system of thermoelasticity is investigated when the initial data have singularities

channel polarization a method for constructing capacity achieving codes for symmetric binary input memoryless channels a method is proposed called channel polarization to construct code sequences that achieve the symmetric capacity i w of any given binary input discrete memoryless channel b dmc w the symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability channel polarization refers to the fact that it is possible to synthesize out of n independent copies of a given b dmc w a second set of n binary input channels wn i 1 les i les n such that as n becomes large the fraction of indices i for which i wn i is near 1 approaches i w and the fraction for which i wn i is near 0 approaches 1 i w the polarized channels wn i are well conditioned for channel coding one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining codes constructed on the basis of this idea are called polar codes the paper proves that given any b dmc w with i w 0 and any target rate r i w there exists a sequence of polar codes cfrn nges1 such that cfrn has block length n 2n rate ges r and probability of block error under successive cancellation decoding bounded as pe n r les o n 1 4 independently of the code rate this performance is achievable by encoders and decoders with complexity o n logn for each