extensions of the method of poles for code construction the method of poles is a method introduced by franaszek 1969 for constructing a rate 1 1 finite state code from k ary data into a constrained channel of finite type whose capacity is strictly greater than log k the method is based on the computation of a set of states called poles with each pole is associated a set of paths going from this pole to others each set verifies an entropy condition the code produced by the method of poles has a sliding block decoder if each set of paths satisfies moreover an optimization condition based on the sum of the path lengths of the set we give a new optimization condition which guarantees the sliding block window decoding property and has a lower computational complexity than the previous one we also extend the method of poles to the more general case of sofic constrained channels

strengthening the gilbert varshamov bound abstract the paper discusses some ways to strengthen nonasymptotically the gilbert varshamov bound for linear codes the unifying idea is to study a certain graph constructed on vectors of low weight in the cosets of the code which we call the varshamov graph various simple estimates of the number of its connected components account for better lower bounds on the minimum distance of codes some of them known in the literature

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

binary construction of quantum codes of minimum distance three and four we give elementary recursive constructions of binary self orthogonal codes with dual distance four for all even lengths n spl ges 12 and n 8 consequently good quantum codes of minimum distance three and four for such length n are obtained via steane s construction and the css construction previously such quantum codes were explicitly constructed only for a sparse set of lengths almost all of our quantum codes of minimum distance three are optimal or near optimal and some of our minimum distance four quantum codes are better than or comparable with those known before

some new classes of quantum mds codes from constacyclic codes quantum maximum distance separable mds codes form an important family of quantum codes in this paper using hermitian construction and classical constacyclic codes we construct six classes of quantum mds codes two of these six classes of quantum mds codes have larger minimum distance than the ones available in the literature most of these quantum mds codes are new in the sense that their parameters are not covered by the codes available in the literature

nonbinary quantum codes we present several results on quantum codes over general alphabets that is in which the fundamental units may have more than two states in particular we consider codes derived from finite symplectic geometry assumed to have additional global symmetries from this standpoint the analogs of calderbank shor steane codes and of gf 4 linear codes turn out to be special cases of the same construction this allows us to construct families of quantum codes from certain codes over number fields in particular we get analogs of quadratic residue codes including a single error correcting code encoding one letter in five for any alphabet size we also consider the problem of fault tolerant computation through such codes generalizing ideas of gottesman see phys rev a vol 57 no 1 p127 37 1998

on a problem concerning the quantum hamming bound for impure quantum codes a famous open problem in the theory of quantum error correcting codes is whether or not the parameters of an impure quantum code can violate the quantum hamming bound for pure quantum codes we partially solve this problem we demonstrate that there exists a threshold n d m such that an arbitrary n k d m quantum code must obey the quantum hamming bound whenever n n d m we list some values of n d m for small d and binary quantum codes

boolean functions projection operators and quantum error correcting codes this paper describes a fundamental correspondence between boolean functions and projection operators in hilbert space the correspondence is widely applicable and it is used in this paper to provide a common mathematical framework for the design of both additive and nonadditive quantum error correcting codes the new framework leads to the construction of a variety of codes including an infinite class of codes that extend the original 5 6 2 code found by rains it also extends to operator quantum error correcting codes

nonbinary stabilizer codes over finite fields one formidable difficulty in quantum communication and computation is to protect information carrying quantum states against undesired interactions with the environment to address this difficulty many good quantum error correcting codes have been derived as binary stabilizer codes fault tolerant quantum computation prompted the study of nonbinary quantum codes but the theory of such codes is not as advanced as that of binary quantum codes this paper describes the basic theory of stabilizer codes over finite fields the relation between stabilizer codes and general quantum codes is clarified by introducing a galois theory for these objects a characterization of nonbinary stabilizer codes over fq in terms of classical codes over fq 2 is provided that generalizes the well known notion of additive codes over f4 of the binary case this paper also derives lower and upper bounds on the minimum distance of stabilizer codes gives several code constructions and derives numerous families of stabilizer codes including quantum hamming codes quadratic residue codes quantum melas codes quantum bose chaudhuri hocquenghem bch codes and quantum character codes the puncturing theory by rains is generalized to additive codes that are not necessarily pure bounds on the maximal length of maximum distance separable stabilizer codes are given a discussion of open problems concludes this paper

asymptotically good quantum codes exceeding the ashikhmin litsyn tsfasman bound it is known that quantum error correction can be achieved using classical binary codes or additive codes over f sub 4 asymptotically good quantum codes have been constructed from algebraic geometry codes and a bound on spl delta r was computed from the tsfasman vladut zink bound of the theory of classical algebraic geometry codes in this correspondence by the use of a concatenation technique we construct a family of asymptotically good quantum codes exceeding the bound in a small interval

quantum reed muller codes a set of quantum error correcting codes based on classical reed muller codes is described the codes have parameters n k d 2 sup r s sup r c r t 2 spl sigma sub i 0 sup t 1 c r i 2 sup t 2 sup t 1

new quantum mds codes from negacyclic codes let q be an odd prime power based on classical negacyclic codes we construct two classes of quantum maximum distance separable mds codes with parameters q2 1 q2 2d 3 d q where q 1 mod 4 and 2 d q 1 is even and q2 1 2 q2 1 2 2d 2 d q where 3 d q is odd some of these quantum mds codes are new in the sense that their parameters are different from all the previously known ones

decoding of dual containing codes from hermitian tower and applications in this paper we study the decoding of dual containing codes from hermitian tower and applications to quantum codes the contribution of this paper is threefold first we construct the quantum stabilizer codes from the hermitian tower second we provide a deterministic decoding algorithm with decoding radius that almost achieves the optimal decoding radius i e 1 r 4 where r is the rate last and most importantly we present a monte carlo algorithm with decoding radius roughly equal to 1 r 3 which is beyond the optimal decoding radius 1 r 4 there are several features in this paper first of all we employ a differential for the hermitian tower this differential plays a crucial role for decoding we also extend our decoding by passing to the constant field extension this constant field extension makes the decoding work perfectly

complexity and sliding block decodability a constrained system or sofic system s is the set of symbol strings generated by the finite length paths through a finite labeled directed graph karabed and marcus 1988 extending the work of adler coppersmith and hassner 1983 used the technique of state splitting to prove the existence of a noncatastrophic rate p q finite state encoder from binary data into s for any input word length p and codeword length q satisfying p q spl les cap s the shannon 1948 capacity for constrained systems that are almost finite type they further proved the existence of encoders enjoying a stronger form of decodability namely sliding block decodability in particular their result implies the existence of a 100 efficient rate 1 2 sliding block code for the charge constrained runlength limited constraint with parameters d k c 1 3 3 an almost finite type system with capacity precisely 1 2 we describe two quite different constructions of such codes the constructions highlight connections between the problem of determining sliding block decodability of a finite state encoder and certain problems of colorability for graphs and sets using these connections we show that the problem of determining the existence of a block decodable input tag assignment for a given rate p q finite state encoder is np complete for p 1 we also prove np completeness results for several related problems in combinatorics and coding

sofic systems and encoding data techniques of symbolic dynamics are applied to prove the existence of codes suitable for certain input restricted channels this generalizes the earlier work of adler coppersmith and hassner on the same problem

nonbinary quantum stabilizer codes we define and show how to construct nonbinary quantum stabilizer codes our approach is based on nonbinary error bases it generalizes the relationship between self orthogonal codes over f sub 4 and binary quantum codes to one between self orthogonal codes over f q sup 2 and q ary quantum codes for any prime power q

enlargement of calderbank shor steane quantum codes it is shown that a classical error correcting code c n k d which contains its dual c sup spl perp spl sube c and which can be enlarged to c n k k 1 d can be converted into a quantum code of parameters n k k n min d 3d 2 this is a generalization of a previous construction it enables many new codes of good efficiency to be discovered examples based on classical bose chaudhuri hocquenghem bch codes are discussed

variable length state splitting with applications to average runlength constrained arc codes a new class of constrained systems average runlength constraints arcs is defined by requiring that the sum of n consecutive run lengths be bounded above by a linear function of n in particular the running average runlength of every sequence in the system is bounded above by a constant a general result is given on the capacity of arc systems the state splitting algorithm is then improved for variable length graphs this is then applied to obtain high fixed rate codes from the free binary source to arc systems as an example a rate 1 2 d k

spatially correlated qubit errors and burst correcting quantum codes we explore the design of quantum error correcting codes for cases where the decoherence events of qubits are correlated in particular we consider the case where only spatially contiguous qubits decohere which is analogous to the case of burst errors in classical coding theory we present several different efficient schemes for constructing families of such codes for example one can find one dimensional quantum codes of length n 13 and 15 that correct burst errors of width b 3 as a comparison a random error correcting quantum code that corrects t 3 errors must have length n spl ges 17 in general we show that it is possible to build quantum burst correcting codes that have near optimal dimension for example we show that for any constant b there exist b burst correcting quantum codes with length n and dimension k n log n o 1 as a comparison the hamming bound for the case with t constant random errors yields k spl les n tlog n o 1

generalized hamming weights of trace codes linear codes over f sub p often admit a natural representation as trace codes of codes that are defined over an extension field f sub p m in the paper the authors obtain estimates for the weights of subcodes of such trace codes their main result is a far reaching generalization of the carlitz uchiyama bound for the duals of binary bch codes in particular they prove sharp bounds for the generalized hamming weights of a large class of codes including duals of bch codes classical goppa codes melas codes and arbitrary cyclic codes of length n p sup m 1 the main tool is the theory of algebraic functions over finite fields in particular the hasse weil bound for the number of places of degree one

pseudo cyclic codes and the construction of quantum mds codes constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable mds codes however the mathematical mechanism may not have been fully understood in this paper we use pseudo cyclic codes which is a further generalization of constacyclic codes to construct the quantum mds codes we can not only provide a unified explanation of many previous constructions but also produce some new quantum mds codes

application of classical hermitian self orthogonal mds codes to quantum mds codes in this paper we first construct several classes of classical hermitian self orthogonal maximum distance separable mds codes through these classical codes we are able to obtain various quantum mds codes it turns out that many of our quantum codes are new in the sense that the parameters of our quantum codes cannot be obtained from all previous constructions

hermitian self dual abelian codes hermitian self dual abelian codes in a group ring fq2 g where fq2 is a finite field of order q2 and g is a finite abelian group are studied using the well known discrete fourier transform decomposition for a semisimple group ring a characterization of hermitian self dual abelian codes in fq2 g is given together with an alternative proof of necessary and sufficient conditions for the existence of such a code in fq2 g i e there exists a hermitian self dual abelian code in fq2 g if and only if the order of g is even and q 2l for some positive integer l later on the study is further restricted to the case where f22l g is a principal ideal group ring or equivalently g a z2k with 2 a based on the characterization obtained the number of hermitian self dual abelian codes in f22l a z2k can be determined easily when a is cyclic this result answers an open problem of jia et al concerning hermitian self dual cyclic codes in many cases f22l a z2k contains a unique hermitian self dual abelian code the criteria for such cases are determined in terms of l and the order of a finally the distribution of finite abelian groups a such that a unique hermitian self dual abelian code exists in f22l a z2 is established together with the distribution of odd integers m such that a unique hermitian self dual cyclic code of length 2 m over f22l exists

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

quantum error detection i statement of the problem this paper is devoted to the problem of error detection with quantum codes we show that it is possible to give a consistent definition of the undetected error event to prove this we examine possible problem settings for quantum error detection our goal is to derive a functional that describes the probability of undetected error under natural physical assumptions concerning transmission with error detection with quantum codes we discuss possible transmission protocols with stabilizer and unrestricted quantum codes the set of results proved in the paper shows that in all the cases considered the average probability of undetected error for a given code is essentially given by one and the same function of its weight enumerators we examine polynomial invariants of quantum codes and show that coefficients of rains s see ibid vol44 p 1388 94 1998 unitary weight enumerators are known for classical codes under the name of binomial moments of the distance distribution as in the classical situation these enumerators provide an alternative expression for the probability of undetected error

classification of q ary perfect quantum codes we solve the problem of the classification of q ary perfect quantum codes we prove that the only nontrivial q ary perfect quantum codes are those with the parameters q2l 1 q2 1 qn 2l 3 q there exist no other nontrivial q ary perfect quantum codes

some good quantum error correcting codes from algebraic geometric codes it is shown that quantum error correction can be achieved by the use of classical binary codes or additive codes over f sub 4 in this correspondence with the help of some algebraic techniques the theory of algebraic geometric codes is used to construct an asymptotically good family of quantum error correcting codes and other classes of good quantum error correcting codes our results are compared with the known best quantum codes

quantum error detection ii bounds for pt i see ibid vol 46 no 3 p 778 88 2000 in part i of this paper we formulated the problem of error detection with quantum codes on the depolarizing channel and gave an expression for the probability of undetected error via the weight enumerators of the code in this part we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent the lower existence bound is proved for stabilizer codes by a counting argument for classical self orthogonal quaternary codes upper bounds are proved by linear programming first we formulate two linear programming problems that are convenient for the analysis of specific short codes next we give a relaxed formulation of the problem in terms of optimization on the cone of polynomials in the krawtchouk basis we present two general solutions of the problem together they give an upper bound on the exponent of undetected error the upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1

bounded delay encodable block decodable codes for constrained systems we introduce and investigate the class of bounded delay encodable block decodable bdb codes several characterizations for this class of codes are given and some construction methods especially for one symbol look ahead bdb codes are described in another direction we use our results to show the existence of a decision procedure for some basic coding problems

polynomial invariants of quantum codes the weight enumerators shor and laflamme 1997 of a quantum code are quite powerful tools for exploring its structure as the weight enumerators are quadratic invariants of the code this suggests the consideration of higher degree polynomial invariants we show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one to one correspondence with s sub k sup n we then present a number of equations and inequalities in these invariants in particular we give a higher order generalization of the shadow enumerator of a code and prove that its coefficients are nonnegative we also prove that the quartic invariants of a 4 4 2 sub 2 code are uniquely determined an important step in a proof that any 4 4 2 sub 2 code is additive rains 1999

on negacyclic mds convolutional codes new families of classical and quantum optimal negacyclic convolutional codes are constructed in this paper this optimality is in the sense that they attain the classical quantum generalized singleton bound the constructions presented in this paper are performed algebraically and not by computational search

a note on the method of poles for code construction the method of poles is a method for constructing a rate 1 1 finite state code from k ary data into a constrained channel s where s is recognized by a given local automaton and s has capacity at least log k we characterize those automata to which the method of poles applies in the case where h s log k the code produced by the method of poles has a sliding block decoder we also give an upper bound on the window length of the decoder that applies when h s spl ges log k

second order asymptotics of sequential hypothesis testing we consider the classical sequential binary hypothesis testing problem in which there are two hypotheses governed respectively by distributions p 0 and p 1 and we would like to decide which hypothesis is true using a sequential test it is known from the work of wald and wolfowitz that as the expectation of the length of the test grows the optimal type i and type ii error exponents approach the relative entropies d p 1 p 0 and d p 0 p 1 we refine this result by considering the optimal backoff or second order asymptotics from the corner point of the achievable exponent region d p 1 p 0 d p 0 p 1 under two different constraints on the length of the test or the sample size first we consider a probabilistic constraint in which the probability that the length of test exceeds a prescribed integer n is less than a certain threshold 0 second the expectation of the sample size is bounded by n in both cases and under mild conditions the second order asymptotics is characterized exactly numerical examples are provided to illustrate our results

on optimal constacyclic codes abstract in this paper we construct new families of maximum distance separable mds convolutional codes derived from classical constacyclic codes additionally we also construct new families of almost mds block and convolutional codes most of the new convolutional codes constructed here are unit memory and have degree 2 moreover based on a different method we construct families of asymmetric quantum codes known in the literature all these results are mainly derived from the investigation of suitable properties on cyclotomic cosets of such codes

sliding block coding for input restricted channels work on coding arbitrary sequences into a constrained system of sequences called a sofic system is presented such systems model the input constraints for input restricted channels e g run length limits and spectral constraints for the magnetic recording channel in this context it is important that the code be noncatastrophic to ensure that the decoder has limited error propagation a constructive proof is given of the existence of finite state invertible noncatastrophic codes from arbitrary n ary sequences to a sofic system s at constant rate p q provided only that shannon s condition p q

monotonicity of the quantum linear programming bound the most powerful technique known at present for bounding the size of quantum codes of prescribed minimum distance is the quantum linear programming bound unlike the classical linear programming bound it is not immediately obvious that if the quantum linear programming constraints are satisfiable for dimension k then the constraints can be satisfied for all lower dimensions we show that the quantum linear programming bound is monotonic in this sense and give an explicitly monotonic reformulation

a linear bound for sliding block decoder window size ii for pt i see ibid vol 34 p 389 99 1988 an input constrained channel is the set s of finite sequences of symbols generated by the walks on a labeled finite directed graph g which is said to present s we introduce a new construction of finite state encoders for input constrained channels the construction is a hybrid of the state splitting technique of adler coppersmith and hassner 1983 and the stethering technique of ashley marcus and roth see ibid vol 41 p 55 76 1995 when s has finite memory and p and g are integers where p g is at most the capacity of s the construction guarantees an encoder at rate p g and having a sliding block decoder literally at rate q p with look ahead that is linear in the number of states of the smallest graph g presenting s this contrasts with previous constructions the straight adler coppersmith and hassner construction provides an encoder having a sliding block decoder at rate q p but the best proven upper bound on the decoder look ahead is exponential in the number of states of g a previous construction of ashley provides an encoder having a sliding block decoder whose look ahead has been proven to be linear in the number of states of g but the decoding is at rate tq tp where t is linear in the number of states of g

a construction of new quantum mds codes it has been a great challenge to construct new quantum maximum distance separable mds codes in particular it is very hard to construct the quantum mds codes with relatively large minimum distance so far except for some sparse lengths all known q ary quantum mds codes have minimum distance q 2 1 in this paper we provide a construction of the quantum mds codes with minimum distance q 2 1 in particular we show the existence of the q ary quantum mds codes with length n q2 1 and minimum distance d for any d q 1 this result extends those given in the works of guardia 2011 jin et al 2010 and kai an zhu 2012 and with length q2 2 3 and minimum distance d for any d 2q 2 3 if 3 q 1 our method is through hermitian selforthogonal codes the main idea of constructing the hermitian self orthogonal codes is based on the solvability in fq of a system of homogenous equations over fq2

css like constructions of asymmetric quantum codes asymmetric quantum error correcting codes aqcs may offer some advantage over their symmetric counterparts by providing better error correction for the more frequent error types the well known css construction of q ary aqcs is extended by removing the fq linearity requirement as well as the limitation on the type of inner product used the proposed constructions are called css like constructions and utilize pairs of nested subfield linear codes under one of the euclidean trace euclidean hermitian and trace hermitian inner products after establishing some theoretical foundations best performing css like aqcs are constructed combining some constructions of nested pairs of classical codes and linear programming many optimal and good pure q ary css like codes for q 2 3 4 5 7 8 9 up to reasonable lengths are found in many instances removing the fq linearity and using alternative inner products give us pure aqcs with improved parameters than relying solely on the standard css construction

on subsystem codes beating the quantum hamming or singleton bound subsystem codes are a generalization of noiseless subsystems decoherence free subspaces and stabilizer codes we generalize the quantum singleton bound to q linear subsystem codes it follows that no subsystem code over a prime field can beat the quantum singleton bound on the other hand we show the remarkable fact that there exist impure subsystem codes beating the quantum hamming bound a number of open problems concern the comparison in the performance of stabilizer and subsystem codes one of the open problems suggested by poulin9s work asks whether a subsystem code can use fewer syndrome measurements than an optimal q linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance we prove that linear subsystem codes cannot offer such an improvement under complete decoding

the method of poles a coding method for constrained channels a method for solving coding problems involving channels restricted by finite state constraints is presented the method was introduced by p a franaszek 1969 and is based on the computation of a set of principal states it is proved that state independent decoding is guaranteed the main result is that the transducer constructed using the method is a local automaton in its output the method is also related to recent work in symbolic dynamics by r l adler 1987 b marcus 1985 and others typical applications are coding methods for magnetic storage

classical and quantum evaluation codes at the trace roots we introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function we give conditions for self orthogonality of these codes and their subfield subcodes with respect to the hermitian inner product they allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature for the binary case we obtain records at http codetables de moreover we obtain several classical linear codes over the field mathbb f 4 which are records at http codetables de

on binary constructions of quantum codes recently steane 1998 proposed an enlargement of the calderbank shor steane construction here we further improve on the parameters obtained from his construction present examples of new codes and analyze asymptotical non constructive bounds

elementary 2 group character codes we describe a class of codes over gf q where q is a power of an odd prime these codes are analogs of the binary reed muller codes and share several features in common with them we determine the minimum weight and properties of these codes for a subclass of codes we find the weight distribution and prove that the minimum nonzero weight codewords give 1 designs

a construction of quantum stabilizer codes based on syndrome assignment by classical parity check matrices in this paper a new but simple construction of stabilizer codes and related entanglement assisted quantum error correcting codes is proposed based on syndrome assignment by classical parity check matrices this method turns the construction of quantum stabilizer codes to the construction of classical parity check matrices satisfying a specific commutative condition the designed minimum distance 2t 1 of the constructed quantum stabilizer codes can be achieved by a commutative classical parity check matrix with classical minimum distance 4t m where the parameter m 0 m 2t depends on a property of the parity check matrix as m decreases there is an increasing set of additional correctable error operators beyond the designed error correcting capability t the asymptotic coding efficiency is at least comparable to that of css codes a class of quantum reed muller codes is constructed and codes in this class have a larger set of correctable error operators than that of the quantum reed muller codes previously developed in the literature quantum circulant codes are also constructed and many of them are optimal in terms of their coding parameters

quantum codes of minimum distance two it is reasonable to expect the theory of quantum codes to be simplified in the case of codes of minimum distance 2 thus it makes sense to examine such codes in the hopes that techniques that prove effective there will generalize with this in mind we present a number of results on codes of minimum distance 2 we first compute the linear programming bound on the dimension of such a code then show that this bound can only be attained when the code either is of even length or is of length 3 or 5 we next consider questions of uniqueness showing that the optimal code of length 2 or 1 is unique implying that the well known one qubit in five single error correcting code is unique and presenting nonadditive optimal codes of all greater even lengths finally we compute the full automorphism group of the more important distance 2 codes allowing us to determine the full automorphism group of any gf 4 linear code

weight formulas for ternary melas codes in this paper we derive a formula for the frequencies of the weights in ternary melas codes and we illustrate this formula by computing a table of examples

beyond stabilizer codes ii clifford codes for pt i see ibid vol 48 no 8 p 2392 95 2002 knill 1996 introduced a generalization of stabilizer codes called clifford codes it remained unclear whether or not clifford codes can be superior to stabilizer codes we show that clifford codes are stabilizer codes provided that the abstract error group has an abelian index group in particular if the errors are modeled by tensor products of pauli matrices then the associated clifford codes are necessarily stabilizer codes

on the construction of nonbinary quantum bch codes four quantum code constructions generating several new families of good nonbinary quantum nonprimitive nonnarrow sense bose chaudhuri hocquenghem codes are presented in this paper the first two are based on calderbank shor steane css construction derived from two nonprimitive bose chaudhuri hocquenghem codes the third one is based on steane s enlargement of nonbinary css codes applied to suitable subfamilies of nonprimitive nonnarrow sense bose chaudhuri hocquenghem codes the fourth construction is derived from suitable subfamilies of hermitian dual containing nonprimitive nonnarrow sense bose chaudhuri hocquenghem codes these constructions generate new families of quantum codes whose parameters are better than the ones available in the literature

clifford code constructions of operator quantum error correcting codes recently operator quantum error correcting codes have been proposed to unify and generalize decoherence free subspaces noiseless subsystems and quantum error correcting codes this correspondence introduces a natural construction of such codes in terms of clifford codes an elegant generalization of stabilizer codes due to knill character theoretic methods are used to derive a simple method to construct operator quantum error correcting codes from any classical additive code over a finite field which obviates the need for self orthogonal codes

on cyclotomic cosets and code constructions abstract new properties of q ary cyclotomic cosets modulo n q m 1 where q 3 is a prime power are investigated in this paper based on these properties the dimension as well as bounds for the designed distance of some families of classical cyclic codes can be computed as an application new families of nonbinary calderbank shor steane css quantum codes as well as new families of convolutional codes are constructed in this work these new css codes have parameters better than the ones available in the literature the convolutional codes constructed here have free distance greater than the ones available in the literature

constacyclic codes and some new quantum mds codes one central theme in quantum error correction is to construct quantum codes that have a large minimum distance quantum maximal distance separable mds codes are optimal in the sense they attain maximal minimum distance recently constructing quantum mds codes has received much attention and seems to become more and more difficult in this paper based on classical constacyclic codes we construct some new quantum mds codes by employing the hermitian construction compared with the known quantum mds codes these quantum mds codes have much larger minimum distance

euclidean and hermitian self orthogonal algebraic geometry codes and their application to quantum codes in the present paper we show that if the dimension of an arbitrary algebraic geometry code over a finite field of even characteristic is slightly less than n 2 g with n being the length of the code and g being the genus of the base curve then it is equivalent to an euclidean self orthogonal code previously such results required a strong condition on the existence of a certain differential we also show a similar result on hermitian self orthogonal algebraic geometry codes as a consequence we can apply our result to quantum codes and obtain some good quantum codes in particular we obtain a q ary quantum q 1 1 mds code for an even power q which is essential for quantum secret sharing

sparse graph codes for quantum error correction sparse graph codes appropriate for use in quantum error correction are presented quantum error correcting codes based on sparse graphs are of interest for three reasons first the best codes currently known for classical channels are based on sparse graphs second sparse graph codes keep the number of quantum interactions associated with the quantum error correction process small a constant number per quantum bit independent of the block length third sparse graph codes often offer great flexibility with respect to block length and rate we believe some of the codes we present are unsurpassed by previously published quantum error correcting codes

on quantum and classical bch codes classical bose chaudhuri hocquenghem bch codes that contain their euclidean or hermitian dual codes can be used to construct quantum stabilizer codes this correspondence studies the properties of such codes it is shown that a bch code of length n can contain its dual code only if its designed distance delta o radicn and the converse is proved in the case of narrow sense codes furthermore the dimension of narrow sense bch codes with small design distance is completely determined and consequently the bounds on their minimum distance are improved these results make it possible to determine the parameters of quantum bch codes in terms of their design parameters

beyond stabilizer codes i nice error bases nice error bases have been introduced by knill 1996 as a generalization of the pauli basis these bases are shown to be projective representations of finite groups we classify all nice error bases of small degree and all nice error bases with abelian index groups we show that in general an index group of a nice error basis is necessarily solvable

quantum shadow enumerators in a previous paper shor and laflamme see phys rev lett vol 78 p 1600 02 1997 define two weight enumerators for quantum error correcting codes connected by a macwilliams 1977 transform and use them to give a linear programming bound for quantum codes we extend their work by introducing another enumerator based on the classical theory of shadow codes that tightens their bounds significantly in particular nearly all of the codes known to be optimal among additive quantum codes codes derived from orthogonal geometry can be shown to be optimal among all quantum codes we also use the shadow machinery to extend a bound on additive codes to general codes obtaining as a consequence that any code of length can correct at most n 1 6 errors

quantum weight enumerators in a recent paper shor and laflamme see phys rev lett vol 78 p 1600 2 1997 defined two weight enumerators for quantum error correcting codes connected by a macwilliams transform and used them to give a linear programming bound for quantum codes we introduce two new enumerators which while much less powerful at producing bounds are useful tools nonetheless the new enumerators are connected by a much simpler duality transform clarifying the duality between shor and laflamme s enumerators we also use the new enumerators to give a simpler condition for a quantum code to have specified minimum distance and to extend the enumerator theory to codes with block size greater than 2

on the classification of hermitian self dual additive codes over rm gf 9 additive codes over gf 9 that are self dual with respect to the hermitian trace inner product have a natural application in quantum information theory where they correspond to ternary quantum error correcting codes however these codes have so far received far less interest from coding theorists than self dual additive codes over gf 4 which correspond to binary quantum codes self dual additive codes over gf 9 have been classified up to length 8 and in this paper we extend the complete classification to codes of lengths 9 and 10 the classification is obtained by using a new algorithm that combines two graph representations of self dual additive codes the search space is first reduced by the fact that every code can be mapped to a weighted graph and a different graph is then introduced that transforms the problem of code equivalence into a problem of graph isomorphism by an extension technique we are able to classify all optimal codes of lengths 11 and 12 there are 56 005 876 11 311 5 codes and 6493 12 312 6 codes we also find the smallest codes with trivial automorphism group

cyclic quantum error correcting codes and quantum shift registers we transfer the concept of linear feedback shift registers to quantum circuits it is shown how to use these quantum linear shift registers for encoding and decoding cyclic quantum error correcting codes

application of constacyclic codes to quantum mds codes quantum maximum distance separable mds codes form an important class of quantum codes to get q ary quantum mds codes one of the effective ways is to find linear mds codes c over f q2 satisfying c h c where c h denotes the hermitian dual code of c for a linear code c of length n over f q2 we say that c is a dual containing code if c h c and c f q2 n several classes of new quantum mds codes with relatively large minimum distance have been produced through dual containing constacyclic mds codes these works motivate us to make a careful study on the existence conditions for dual containing constacyclic codes we obtain necessary and sufficient conditions for the existence of dual containing constacyclic codes four classes of dual containing constacyclic mds codes are constructed and their parameters are computed consequently the quantum mds codes are derived from these parameters the quantum mds codes exhibited here have minimum distance bigger than the ones available in the literature

on the construction of bounded delay encodable codes for constrained systems we present a new technique to construct sliding block modulation codes with a small decoding window our method which involves both state splitting and look ahead encoding crucially depends on a new local construction method for bounded delay codes we apply our method to construct several new codes all with a smaller decoding window than previously known codes for the same constraints at the same rate

some new constructions of quantum mds codes it is an important task to construct quantum maximum distance separable mds codes with good parameters in the present paper we provide six new classes of q ary quantum mds codes by using generalized reed solomon grs codes and hermitian construction the minimum distances of our quantum mds codes can be larger than frac q 2 1 three of these six classes of quantum mds codes have longer lengths than the ones constructed in 1 and 2 hence some of their results can be easily derived from ours via the propagation rule moreover some known quantum mds codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum mds codes are also improved as well

quantum codes 6 2 3 sub p and 7 3 3 sub p p spl ges 3 exist we prove that nonbinary quantum stabilizer codes with parameters n k d sub p 6 2 3 sub p and 7 3 3 sub p exist for all odd primes p by using graph machinery given by schlingemann and werner see phys rev a vol 65 no 012308 2001 with a little number theory and combinatorics

a finite gilbert varshamov bound for pure stabilizer quantum codes a finite gilbert varshamov gv bound for pure stabilizer binary and nonbinary quantum error correcting codes is presented in analogy to the gv bound for classical codes by using several enumerative results in finite unitary geometry from this quantum gv bound we obtain several new binary quantum codes in a nonconstructive way having better parameters than the known codes

euclidean and hermitian hulls of mds codes and their applications to eaqeccs in this paper we construct several classes of maximum distance separable mds codes via generalized reed solomon grs codes and extended grs codes where we can determine the dimensions of their euclidean hulls or hermitian hulls it turns out that the dimensions of euclidean hulls or hermitian hulls of the codes in our constructions can take all or almost all possible values as a consequence we can apply our results to entanglement assisted quantum error correcting codes eaqeccs and obtain several new families of mds eaqeccs with flexible parameters the required number of maximally entangled states of these mds eaqeccs can take all or almost all possible values moreover several new classes of q ary mds eaqeccs of length n q 1 are also obtained

linear programming bounds for entanglement assisted quantum error correcting codes by split weight enumerators linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes in this paper we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator after deriving the macwilliams identities for these enumerators we are able to prove algebraic linear programming bounds such as the singleton bound the hamming bound and the first linear programming bound our singleton bound and hamming bound are more general than the previous bounds for entanglement assisted quantum stabilizer codes in addition we show that the first linear programming bound improves the hamming bound when the relative distance is sufficiently large on the other hand we obtain additional constraints on the size of pauli subgroups for quantum codes which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length in particular we show that there is no 27 15 5 or 28 14 6 stabilizer code we also discuss the existence of some entanglement assisted quantum stabilizer codes with maximal entanglement as a result the upper and lower bounds on the minimum distance of maximal entanglement quantum stabilizer codes with length up to 20 are significantly improved