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IEEE Transactions on Information Theory

Generalized Bose–Lin Codes, a Class of Codes Detecting Asymmetric Errors

Irina Naydenova, Torleiv Kløve

Published Mar 1, 2007• 10 citations
Publisher

Abstract

Bose and Lin introduced a class of systematic codes for detection of binary asymmetric errors. In this note, we describe a generalization to q-ary asymmetric error detecting codes. For these codes, the possible undetectable errors are characterized and the undetectable errors of minimum weight are determined
DOI
10.1109/tit.2006.890776

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We study the error detection problem in q-ary asymmetric channels, wherein every input symbol xiis mapped to an output symbol yisatisfying yi≥xi. A general setting is assumed, where the noise vectors are (potentially) restricted in: 1) the amplitude, yi-xi≤a, 2) the Hamming weight, Σi=1n1{yi≠xi}≤ h, and 3) the total weight, Σi=1n(yi-xi)≤t. Optimal codes detecting these types of errors are described for certain sets of parameters a, h, t, both in the standard and in the cyclic (mod q) version of the problem. It is also demonstrated that these codes are optimal in the large alphabet limit for every a, h, t and every block-length n.
In this paper, we present a characterization of (binary and non-binary) quantum error-correcting codes. Based on this characterization, we introduce a method to construct p p -ary quantum codes using Boolean functions satisfying a system of certain quadratic relations. As a consequence of the construction, we are able to construct quantum codes of minimum distance 2 2 . In particular, we produce a class of binary quantum ( ( n , 2 n − 2 − 1 2 ( n − 1 ( n − 1 ) / 2 ) , 2 ) ) ((n,2^{n-2}-\frac 12{n-1\choose (n-1)/2},2)) -codes for odd length n ≥ 5 n\ge 5 . For n ≥ 11 n\ge 11 , this improves the result by Rains in Quantum codes of minimal distance two , 1999, showing the existence of binary quantum ( ( n , 3 ⋅ 2 n − 4 , 2 ) ) ((n,3\cdot 2^{n-4},2)) -codes for odd n ≥ 5 n\ge 5 . Moreover, our binary quantum
There are two basic methods of error control for communication, both involving coding of the messages. With forward error correction, the codes are used to detect and correct errors. In a repeat request system, the codes are used to detect errors and, if there are errors, request a retransmission. Error detection is usually much simpler to implement than error correction and is widely used. However, it is given a very cursory treatment in almost all textbooks on coding theory. Only a few older books are devoted to error detecting codes. This book begins with a short introduction to the theory of block codes with emphasis on the parts important for error detection. The weight distribution is particularly important for this application and is treated in more detail than in most books on error correction. A detailed account of the known results on the probability of undetected error on the q-ary symmetric channel is also given.
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.