Sufficient Conditions for Monotonicity of the Undetected Error Probability for Large Channel Error Probabilities

The performance of a linear error-detecting code in a symmetric memoryless channel is characterized by its probability of undetected error, which is a function of the channel symbol error probability, involving basic parameters of a code and its weight distribution. However, the code weight distribution is known for relatively few codes since its computation is an NP-hard problem. It should therefore be useful to have criteria for properness and goodness in error detection that do not involve the code weight distribution. In this work we give two such criteria. We show that a binary linear code C of length n and its dual code C^⊥ of minimum code distance d^⊥ are proper for error detection whenever d^⊥ ≥ ?n/2? + 1, and that C is proper in the interval [(n + 1 ? 2d^⊥)/(n ? d^⊥); 1/2] whenever ?n/3? + 1 ≤ d^⊥ ≤ ?n/2?. We also provide examples, mostly of Griesmer codes and their duals, that satisfy the above conditions.     

Extended Binomial Moments of a Linear Code and the Undetected Error Probability

Extended binomial moments of a linear code, introduced in this paper, are synonymously related to the code weight distribution and linearly to its binomial moments. In contrast to the latter, the extended binomial moments are monotone, which makes them appropriate for studying the undetected error probability. We establish some properties of the extended binomial moments and, based on this, derive new lower and upper bounds on the probability of undetected error. Also, we give a simplification of some previously obtained sufficient conditions for proper and good codes, stated in terms of the extended binomial moments.     

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 error-detecting performance of some classes of block codes

We establish the properness of some classes of binary block codes with symmetric distance distribution, including Kerdock codes and codes that satisfy the Grey-Rankin bound, as well as the properness of Preparata codes, thus augmenting the list of very few known proper nonlinear codes.Translated from Problemy Peredachi Informatsii, No. 4, 2004, pp. 68–78. Original Russian Text Copyright © 2004 by Dodunekova, Dodunekov, Nikolova.     

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.