Rossitza Dodunekova

Limit theorems for the Petersburg game

We determine all possible subsequences \(\left\{ {n_k } \right\}_{k = 1}^\infty\)of the positive integers for which the suitably centered and normalized total gain S nk in n k Petersburg games has an asymptotic distribution as k → ∞, and identify the corresponding set of Hmiting distributions. We also solve all the companion problems for lightly, moderately, and heavily trimmed versions of the sum S nk and for the respective sums of extreme values in S nk .

Sufficient conditions for good and proper error-detecting codes

The performance of linear block codes over a finite field is investigated when they are used for pure error detection. Sufficient conditions for a code to be good or proper for error detection are derived.

A survey on proper codes

The performance of a linear t-error correcting code over a q-ary symmetric memoryless channel with symbol error probability @e is characterized by the probability that a transmission error will remain undetected. This probability is a function of @e involving the code weight distribution and the weight distribution of the cosets of minimum weight at most t. When the undetectable error probability is an increasing function of @e, the code is called t-proper. The paper presents sufficient conditions for t-properness and a list of codes known to be proper, many of which have been studied by these sufficient conditions. Special attention is paid to error detecting codes of interest in modern communication.

Almost-MDS and near-MDS codes for error detection

The error detection capability of almost-MDS (AMDS) and nearly-MDS (NMDS) codes is studied. Necessary and sufficient conditions for the codes to be proper or good for error detection are given.

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.

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, MacDonalds 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.

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.

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.

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.

The domain of partial attraction of a poisson law

Groshev gave a characterization of the union of domains of partial attraction of all Poisson laws in 1941. His classical condition is expressed by the underlying distribution function and disguises the role of the mean λ of the attracting distribution. In the present paper we start out from results of the recent “probabilistic approach” and derive characterizations for any fixed λ>0 in terms of the underlying quantile function. The approach identifies the portion of the sample that contributes the limiting Poisson behavior of the sum, delineates the effect of extreme values, and leads to necessary and sufficient conditions all involving λ. It turns out that the limiting Poisson distributions arise in two qualitatively different ways depending upon whether λ>1 or λ<1. A concrete construction, illustrating all the results, also shows that in the boundary case when λ=1 both possibilities may occur.