A generalization of bounds for cyclic codes, including the HT and BS bounds

We use the algebraic structure of cyclic codes and some properties of the discrete Fourier transform to give a reformulation of several classical bounds for the distance of cyclic codes, by extending techniques of linear algebra. We propose a bound, whose computational complexity is polynomial bounded, which is a generalization of the Hartmann-Tzeng bound and the Betti-Sala bound. In the majority of computed cases, our bound is the tightest among all known polynomial-time bounds, including the Roos bound

On the Triple-Error-Correcting Cyclic Codes with Zero Set {1,2 i + 1,2 j + 1}

Abstract. Weconsideraclassof3-error-correctingcycliccodesoflength 2 m −1 over the two-element field F2. The generator polynomial of a code of this class has zeroes α,α 2i +1 and α 2 j +1, where α is a primitive element of the field F2 m. In short, {1,2i +1,2 j +1} refers to the zero set of these codes. Kasami in 1971 and Bracken and Helleseth in 2009, showed that cyclic codes with zeroes {1,2 ℓ +1,2 3ℓ +1} and {1,2 ℓ +1,2 2ℓ +1} respectively are 3-error correcting, where gcd(ℓ,m) = 1. We present a sufficient condition so that the zero set {1,2 ℓ +1,2 pℓ +1}, gcd(ℓ,m) = 1 gives a 3-error-correcting cyclic code. The question for p> 3 is open. In addition, we determine all the 3-error-correcting cyclic codes in the class {1,2 i + 1,2 j + 1} for m < 20. We investigate their weight distribution via their duals and observe that they have the same weight distribution as 3-error-correcting BCH codes for m < 14. Further our experiment shows that these codes are not equivalent to the 3-error-correcting BCH code in general. We also study the Schaub algorithm which determines a lower bound of the minimum distance of a cyclic code. We introduce a pruning strategy to improve the Schaub algorithm. Finally we study the cryptographic property of a Boolean function, called spectral immunity which is directly related to the minimum distance of cyclic codes over F2m. We apply the improved Schaub algorithm in order to find a lower bound of the spectral immunity of a Boolean function related to the zero set {1,2 i +1,2 j +1}

Cellular Tree Classifiers

The cellular tree classifier model addresses a fundamental problem in the design of classifiers for a parallel or distributed computing world: Given a data set, is it sufficient to apply a majority rule for classification, or shall one split the data into two or more parts and send each part to a potentially different computer (or cell) for further processing? At first sight, it seems impossible to define with this paradigm a consistent classifier as no cell knows the “original data size”, n. However, we show that this is not so by exhibiting two different consistent classifiers. The consistency is universal but is only shown for distributions with nonatomic marginals. Index Terms — Classification, pattern recognition, tree classifiers, cellular computation, Bayes risk consistency, asymptotic analysis, nonparametric estimation