We derive new upper bounds on the error exponents for the maximum likelihood decoding and error detecting in the binary symmetric channels. This is an improvement on the straight-line bound by Shannon-Gallager-Berlekamp (1967) and the McEliece-Omura (1977) minimum distance bound. For the probability of undetected error the new bounds are better than the recent bound by Abdel-Ghaffar (1997) and the minimum distance and straight-line bounds by Levenshtein (1978, 1989). We further extend the range of rates where the undetected error exponent is known to be exact. Keywords: Error exponents, Undetected error, Maximum likelihood decoding, Distance distribution, Krawtchouk polynomials. Submitted to IEEE Transactions on Information Theory 1 Introduction A classical problem of the information theory is to estimate probabilities of undetected and decoding errors when a block code is used for information transmission over a binary symmetric channel (BSC). We will study here exponential bounds ...

Abstract — This paper is concerned with bounds for quantum error-correcting codes. Using the quantum MacWilliams identities, we generalize the linear programming approach from classical coding theory to the quantum case. Using this approach, we obtain Singleton-type, Hamming-type, and the first linearprogramming-type bounds for quantum codes. Using the special structure of linear quantum codes, we derive an upper bound that is better than both Hamming and the first linear programming bounds on some subinterval of rates. Index Terms — Linear programming, quantum codes, upper bounds.

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Abstract—We consider the problem of bounding the distance distribution for unrestricted block codes with known distance and/or dual distance. Applying the polynomial method, we provide a general framework for previously known results. We derive several upper and lower bounds both for finite length and for sequences of codes of growing length. Asymptotic results in the paper improve previously known estimates. In particular, we prove the best known bounds on the binomiality range of the distance spectrum of codes with a known dual distance. Index Terms—Binomial spectrum, constant weight codes, distance distribution, Krawtchouk polynomials, polynomial method. I.

Abstract We are interested in the maximal size A(n, d) of a binary error-correcting code of length n and distance d, or, alternatively, in the best packing of balls of radius (d- 1)/2 in the n-dimensional Hamming space.The best known lower bound on A(n, d) is due to Gilbert and Varshamov, and is obtainedby a covering argument. The best know upper bound is due to McEliece, Rodemich, Rumsey and Welch, and is obtained using Delsarte's linear programming approach. It is not known,whether this is the best possible bound one can obtain from Delsarte's linear program. We show that the optimal upper bound obtainable from Delsarte's linear program willstrictly exceed the Gilbert-Varshamov lower bound. In fact, it will be at least as big as the average of the Gilbert-Varshamov bound and the McEliece, Rodemich, Rumsey and Welchupper bound. Similar results hold for constant weight binary codes.The average of the Gilbert-Varshamov bound and the McEliece, Rodemich, Rumsey and Welch upper bound might be the true value of Delsarte's bound. We provide some evidencefor this conjecture.

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 ...

We derive a new upper bound on the exponent of error probability of decoding for the best possible codes in the Gaussian channel. This bound is tighter than the known upper bounds (the sphere-packing and minimum-distance bounds proved in Shannon’s classical 1959 paper and their low-rate improvement by Kabatiansky and Levenshtein). The proof is accomplished by studying asymptotic properties of codes on the sphere I @ A. First we prove a general lower bound on the distance distribution of codes of large size. To derive specific estimates of the distance distribution, we study the asymptotic behavior of Jacobi polynomials as. Since on the average there are many code vectors in the vicinity of the transmitted vector, one can show that the probability of confusing and one of these vectors cannot be too small. This proves a lower bound on the error probability of decoding and the upper bound announced in the title.

The covering radius of a code tells us how far in the sense of Hamming distance an arbitrary word of the ambient space can be from the code. For a few decades this parameter has been widely studied. In this paper we estimate the covering radius of a code when the dual distance is known. We derive a new bound on covering radii of linear codes. It improves essentially on the previously known estimates in a certain wide range. We also study asymptotic bounds on the cardinality of constant weight codes. Index Terms: Covering radius, dual distance, Krawtchouk polynomial, constant weight codes. 1

2. Suppose the smallest Hamming weight of non-zero vectors in V is d. (In coding-theoretic terminology, V is a linear code of length n, rate r and distance d.) We settle two extremal problems on such spaces. First, we prove a (weak form) of a conjecture by Kalai and Linial and show that the fraction of vectors in V with weight d is exponentially small. Specifically, in the interesting case of a small r, this fraction does not exceed 2 \Gamma \Omega ( r 2 log(1=r)+1 n). We also answer a question of Ben-Or and show that if r? 1

Abstract We prove two results about the value of Delsarte's linear program for binary codes.Our main result is a new lower bound on the value of the program, which, in particular, is nearly tight for low rate codes.We also give an easy proof of an upper bound, which coincides with the best known bound for a wide range of parameters. 0