Evaluation of union bounds for space-time codes based on a common function

Error-rate evaluation of Space-Time codes using Union bounds sometimes requires very heavy computational loads and so is impractical to use. In this paper, a Common function shared by different Union bounds is derived and used to develop a modified Union bound (MUB) for error-rate evaluation. Results of numerical evaluations and Monte-Carlo simulation on two 2x2 rotation-based S-T codes show that the MUB provides a good compromise between the required computational load and the accuracy for error-rate evaluation. ©2009 IEEE.published_or_final_versionThe 22nd IEEE Canadian Conference on Electrical and Computer Engineering (CCECE 2009), St. John's, NL., 3-6 May 2009. In Proceedings of the 22nd CCECE, 2009, p. 89-9

Reed-Muller codes for random erasures and errors

This paper studies the parameters for which Reed-Muller (RM) codes over $GF(2)$ can correct random erasures and random errors with high probability, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate $GF(2)$ polynomials on random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity. The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about $E(m,r)$, the matrix whose rows are truth tables of all monomials of degree $\leq r$ in $m$ variables. What is the most (resp. least) number of random columns in $E(m,r)$ that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees $r$, which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code $C$ of sufficiently high rate we construct a new code $C'$, also of very high rate, such that for every subset $S$ of coordinates, if $C$ can recover from erasures in $S$, then $C'$ can recover from errors in $S$. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent \cite{KLP} bounds from constant degree to linear degree polynomials

SmartUnit: Empirical Evaluations for Automated Unit Testing of Embedded Software in Industry

In this paper, we aim at the automated unit coverage-based testing for embedded software. To achieve the goal, by analyzing the industrial requirements and our previous work on automated unit testing tool CAUT, we rebuild a new tool, SmartUnit, to solve the engineering requirements that take place in our partner companies. SmartUnit is a dynamic symbolic execution implementation, which supports statement, branch, boundary value and MC/DC coverage. SmartUnit has been used to test more than one million lines of code in real projects. For confidentiality motives, we select three in-house real projects for the empirical evaluations. We also carry out our evaluations on two open source database projects, SQLite and PostgreSQL, to test the scalability of our tool since the scale of the embedded software project is mostly not large, 5K-50K lines of code on average. From our experimental results, in general, more than 90% of functions in commercial embedded software achieve 100% statement, branch, MC/DC coverage, more than 80% of functions in SQLite achieve 100% MC/DC coverage, and more than 60% of functions in PostgreSQL achieve 100% MC/DC coverage. Moreover, SmartUnit is able to find the runtime exceptions at the unit testing level. We also have reported exceptions like array index out of bounds and divided-by-zero in SQLite. Furthermore, we analyze the reasons of low coverage in automated unit testing in our setting and give a survey on the situation of manual unit testing with respect to automated unit testing in industry.Comment: In Proceedings of 40th International Conference on Software Engineering: Software Engineering in Practice Track, Gothenburg, Sweden, May 27-June 3, 2018 (ICSE-SEIP '18), 10 page

Finite-state codes

A class of codes called finite-state (FS) codes is defined and investigated. The codes, which generalize both block and convolutional codes, are defined by their encoders, which are finite-state machines with parallel inputs and outputs. A family of upper bounds on the free distance of a given FS code is derived. A general construction for FS codes is given, and it is shown that in many cases the FS codes constructed in this way have a free distance that is the largest possible. Catastrophic error propagation (CEP) for FS codes is also discussed. It is found that to avoid CEP one must solve the graph-theoretic problem of finding a uniquely decodable edge labeling of the state diagram

Coding for Parallel Channels: Gallager Bounds for Binary Linear Codes with Applications to Repeat-Accumulate Codes and Variations

This paper is focused on the performance analysis of binary linear block codes (or ensembles) whose transmission takes place over independent and memoryless parallel channels. New upper bounds on the maximum-likelihood (ML) decoding error probability are derived. These bounds are applied to various ensembles of turbo-like codes, focusing especially on repeat-accumulate codes and their recent variations which possess low encoding and decoding complexity and exhibit remarkable performance under iterative decoding. The framework of the second version of the Duman and Salehi (DS2) bounds is generalized to the case of parallel channels, along with the derivation of their optimized tilting measures. The connection between the generalized DS2 and the 1961 Gallager bounds, addressed by Divsalar and by Sason and Shamai for a single channel, is explored in the case of an arbitrary number of independent parallel channels. The generalization of the DS2 bound for parallel channels enables to re-derive specific bounds which were originally derived by Liu et al. as special cases of the Gallager bound. In the asymptotic case where we let the block length tend to infinity, the new bounds are used to obtain improved inner bounds on the attainable channel regions under ML decoding. The tightness of the new bounds for independent parallel channels is exemplified for structured ensembles of turbo-like codes. The improved bounds with their optimized tilting measures show, irrespectively of the block length of the codes, an improvement over the union bound and other previously reported bounds for independent parallel channels; this improvement is especially pronounced for moderate to large block lengths.Comment: Submitted to IEEE Trans. on Information Theory, June 2006 (57 pages, 9 figures