Covering radius and dual distance
Designs, Codes and Cryptography
Error control systems for digital communication and storage
Error control systems for digital communication and storage
Bounds on Spectra of Codes with Known Dual Distance
Designs, Codes and Cryptography
ACS'07 Proceedings of the 7th Conference on 7th WSEAS International Conference on Applied Computer Science - Volume 7
Estimates for the range of binomiality in codes' spectra
IEEE Transactions on Information Theory
Linear programming bounds for doubly-even self-dual codes
IEEE Transactions on Information Theory
On relations between covering radius and dual distance
IEEE Transactions on Information Theory
Estimates of the distance distribution of codes and designs
IEEE Transactions on Information Theory
Bounds on distance distributions in codes of known size
IEEE Transactions on Information Theory
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Dual codes play an important role in the field of error detecting codes on a binary symmetric channel. Via the MacWilliams Identities they can be used to calculate the original code's weight distribution and its probability of undetected error. Moreover, knowledge of the minimum distance of the dual code provides insight in the properties of the weights of a code. In this paper firstly the order of growth of the dual distance of a CRC as a function of the block length n is investigated, and a new lower bound is proven. Then this bound is used to derive a weaker version of the 2-r-bound on the probability of undetected error, and the relationship of this bound to the 2-r-bound is discussed. Estimates of the range of binomiality and the covering radius are given, depending only on the code rate R and the degree r of the generating polynomial of the CRC. In the case of a CRC, two results of Tietäväinen are improved. Furthermore, wit is shown that there is binomial behavior of the weight distribution, if only n is large enough. Then, by means of an estimate of the tail of the binomial, another bound on the probability of undetected error is verified. Finally a new version of Sidel'nikov's theorem on the normality of the cumulative distribution function of the weights of a code is presented, where the dual distance is replaced by an expression depending on n and the degree r. In this way the conclusions of the present paper may attribute a new meaning to some well known results about codes with known dual distance and give some new insight in this kind of problems.