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     

Undetected error probabilities of binary primitive BCH codes forboth error correction and detection

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

On the Probability of Undetected Error for Linear Block Codes

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

The MMD codes are proper for error detection

The undetected error probability of a linear code used to detect errors on a symmetric channel is a function of the symbol error probability /spl epsi/ of the channel and involves the weight distribution of the code. The code is proper, if the undetected error probability increases monotonously in /spl epsi/. Proper codes are generally considered to perform well in error detection. We show in this correspondence that maximum minimum distance (MMD) codes are proper.     

Binomial moments of the distance distribution: bounds andapplications

We study a combinatorial invariant of codes which counts the number of ordered pairs of codewords in all subcodes of restricted support in a code. This invariant can be expressed as a linear form of the components of the distance distribution of the code with binomial numbers as coefficients. For this reason we call it a binomial moment of the distance distribution. Binomial moments appear in the proof of the MacWilliams (1963) identities and in many other problems of combinatorial coding theory. We introduce a linear programming problem for bounding these linear forms from below. It turns out that some known codes (1-error-correcting perfect codes, Golay codes, Nordstrom-Robinson code, etc.) yield optimal solutions of this problem, i.e., have minimal possible binomial moments of the distance distribution. We derive several general feasible solutions of this problem, which give lower bounds on the binomial moments of codes with given parameters, and derive the corresponding asymptotic bounds. Applications of these bounds include new lower bounds on the probability of undetected error for binary codes used over the binary-symmetric channel with crossover probability p and optimality of many codes for error detection. Asymptotic analysis of the bounds enables us to extend the range of code rates in which the upper bound on the undetected error exponent is tight     

On the undetected error probability for binary codes

In this paper, the undetected error probability for binary codes is studied. First complementary codes are studied. Next, a new proof of Abdel-Ghaffar's (1997) lower bound on the undetected error probability is presented and some generalizations are given. Further, upper and lower bounds on the undetected error probability for binary constant weight codes are given, and asymptotic versions are studied.     

A lower bound on the undetected error probability and strictlyoptimal codes

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

On the Undetected Error Probability for Shortened Hamming Codes

Shortened Hamming codes are widely used for error detection in data communications. In this paper, a method for computing the probability of an undetected error for these codes is presented. This method is then used to evaluate the error-detection performance of the shortened codes obtained from the two distance-4 Hamming codes adopted by CCITT X.25 for error control for packet-switched networks. We show that shortening a code does affect its error-detection performance.     

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     

The duals of MMD codes are proper for error detection

A linear code, when used for error detection on a symmetric channel, is said to be proper if the corresponding undetected error probability increases monotonically in /spl epsiv/, the symbol error probability of the channel. Such codes are generally considered to perform well in error detection. A number of well-known classes of linear codes are proper, e.g., the perfect codes, MDS codes, MacDonald's codes, MMD codes, and some Near-MDS codes. The aim of this work is to show that also the duals of MMD codes are proper.     

On the Probability of Undetected Error for the Maximum Distance Separable Codes

In this paper we investigate the performance of maximum-distance-separable codes with symbols fromGF(q)when they are used for pure error detection or for simultaneous error correction and detection over aq-input andq-output discret memoryless channel with symbol error probability ε. First we show that the probability of undetected error for an MDS code used for pure error detection is upper bounded byq^{-r}and decreases monotonically as εdecreases from(q - 1)/qto 0, whereris the number of parity-check symbols of the code. Then we show that the probability of undetected error for an MDS code used for correctingtor fewer symbol errors is upper bounded byq^{-r} Summin{i=0}max{t}(min{i} max{n})(q - 1)^{i}and decreases monotonically as ε decreases from(q - 1)/qto 0. These results show that the MDS codes are effective for both pure error detection and simultaneous error correction and detection.     

Necessary conditions for good error detection

The problem of determining the error detection capabilities of linear codes where the channel is a binary symmetric channel is addressed. Necessary conditions are given on the number of codewords in an [n,k] linear code and its dual for the probability of an undetected error to be upper bounded by 2^-(n-k)     

Concerning a bound on undetected error probability (Corresp.)

In the past, it has generally been assumed that the probability of undetected error for an(n,k)block code, used solely for error detection on a binary symmetric channel, is upperbounded by2^{-(n-k)}. In this correspondence, it is shown that Hamming codes do indeed obey this bound, but that the bound is violated by some more general codes. Examples of linear, cyclic, and Bose-Chaudhuri-Hocquenghem (BCH) codes which do not obey the bound are given.     

Bounds on distance distributions in codes of known size

We treat the problem of bounding components of the possible distance distributions of codes given the knowledge of their size and possibly minimum distance. Using the Beckner inequality from harmonic analysis, we derive upper bounds on distance distribution components which are sometimes better than earlier ones due to Ashikhmin, Barg, and Litsyn. We use an alternative approach to derive upper bounds on distance distributions in linear codes. As an application of the suggested estimates we get an upper bound on the undetected error probability for an arbitrary code of given size. We also use the new bounds to derive better upper estimates on the covering radius, as well as a lower bound on the error-probability threshold, as a function of the code's size and minimum distance.     

LDPC码的不可检测译码错误分析

纠错性能良好的LDPC码同时具有良好的检错性能。本文介绍了LDPC码的不可检测译码错误概率上限，讨论了不可检测译码错误与码参数的关系，提出对于特定构造的LDPC码，在基于AWGN信道传输和置信传播迭代译码时，可以应用迭代过程中连续消息的密度进化来分析不可检测译码错误概率，并给出了一些短码长高码率情况下的仿真结果。     

Design and analysis of codes with distance 4 and 6 minimizing the probability of decoder error

The problem of minimization of the decoder error probability is considered for shortened codes of dimension 2 ^m with distance 4 and 6. We prove that shortened Panchenko codes with distance 4 achieve the minimal probability of decoder error under special form of shortening. This shows that Hamming codes are not the best. In the paper, the rules for shortening Panchenko codes are defined and a combinatorial method to minimize the number of words of weight 4 and 5 is developed. There are obtained exact lower bounds on the probability of decoder error and the full solution of the problem of minimization of the decoder error probability for [39,32,4] and [72,64,4] codes. For shortened BCH codes with distance 6, upper and lower bounds on the number of minimal weight codewords are derived. There are constructed [45,32,6] and [79,64,6] BCH codes with the number of weight 6 codewords close to the lower bound and the decoder error probabilities are calculated for these codes. The results are intended for use in memory devices.     

The worst case probability of undetected error for linear codes onthe local binomial channel

The worst case probability of undetected error for a linear [n,k:q] code used on a local binomial channel is studied. For the two most important cases it is determined in terms of the weight hierarchy of the code. The worst case probability of undetected error is determined explicitly for some classes of codes     

A low-rate bound on the reliability of a quantum discrete memoryless channel

We extend a low-rate improvement of the random coding bound on the reliability of a classical discrete memoryless channel (DMC) to its quantum counterpart. The key observation that we make is that the problem of bounding below the error exponent for a quantum channel relying on the class of stabilizer codes is equivalent to the problem of deriving error exponents for a certain symmetric classical channel.     

The Higher Weight Enumerators of the Doubly-Even, Self-Dual [48, 24, 12] Code

An efficient algorithm is presented for calculating higher weight enumerators of linear codes given generator matrices. By this algorithm, the higher weight enumerators of the unique doubly-even, self-dual code are calculated. The algorithm is based on a previously shown relationship between Tutte polynomials and higher weight enumerators.     

Accumulate codes based on 1+D convolutional outer codes

A new construction of good, easily encodable, and soft-decodable codes is proposed in this paper. The construction is based on serially concatenating several simple 1+D convolutional codes as the outer code, and a rate-1 1/(1+D) accumulate code as the inner code. These codes have very low encoding complexity and require only one shift-forward register for each encoding branch. The input-output weight enumerators of these codes are also derived. Divsalar?s simple bound technique is applied to analyze the bit error rate performance, and to assess the minimal required signal-to-noise ratio (SNR) for these codes to achieve reliable communication under AWGN channel. Simulation results show that the proposed codes can provide good performance under iterative decoding.