Error Patterns

In coding theory the problem of decoding focuses on error vectors. In the simplest situation code words are $(0,1)$-vectors, as are the received messages and the error vectors. Comparison of a received word with the code words yields a set of error vectors. In deciding on the original code word, usually the one for which the error vector has minimum Hamming weight is chosen. In this note some remarks are made on the problem of the elements 1 in the error vector, that may enable unique decoding, in case two or more code words have the same Hamming distance to the received message word, thus turning error detection into error correction. The essentially new aspect is that code words, message words and error vectors are put in one-one correspondence with graphs

Convolutional-Code-Specific CRC Code Design

Cyclic redundancy check (CRC) codes check if a codeword is correctly received. This paper presents an algorithm to design CRC codes that are optimized for the code-specific error behavior of a specified feedforward convolutional code. The algorithm utilizes two distinct approaches to computing undetected error probability of a CRC code used with a specific convolutional code. The first approach enumerates the error patterns of the convolutional code and tests if each of them is detectable. The second approach reduces complexity significantly by exploiting the equivalence of the undetected error probability to the frame error rate of an equivalent catastrophic convolutional code. The error events of the equivalent convolutional code are exactly the undetectable errors for the original concatenation of CRC and convolutional codes. This simplifies the computation because error patterns do not need to be individually checked for detectability. As an example, we optimize CRC codes for a commonly used 64-state convolutional code for information length k=1024 demonstrating significant reduction in undetected error probability compared to the existing CRC codes with the same degrees. For a fixed target undetected error probability, the optimized CRC codes typically require 2 fewer bits.Comment: 12 pages, 8 figures, journal pape

A study of pattern recovery in recurrent correlation associative memories

In this paper, we analyze the recurrent correlation associative memory (RCAM) model of Chiueh and Goodman. This is an associative memory in which stored binary memory patterns are recalled via an iterative update rule. The update of the individual pattern-bits is controlled by an excitation function, which takes as its arguement the inner product between the stored memory patterns and the input patterns. Our contribution is to analyze the dynamics of pattern recall when the input patterns are corrupted by noise of a relatively unrestricted class. We make three contributions. First, we show how to identify the excitation function which maximizes the separation (the Fisher discriminant) between the uncorrupted realization of the noisy input pattern and the remaining patterns residing in the memory. Moreover, we show that the excitation function which gives maximum separation is exponential when the input bit-errors follow a binomial distribution. Our second contribution is to develop an expression for the expectation value of bit-error probability on the input pattern after one iteration. We show how to identify the excitation function which minimizes the bit-error probability. However, there is no closed-form solution and the excitation function must be recovered numerically. The relationship between the excitation functions which result from the two different approaches is examined for a binomial distribution of bit-errors. The final contribution is to develop a semiempirical approach to the modeling of the dynamics of the RCAM. This provides us with a numerical means of predicting the recall error rate of the memory. It also allows us to develop an expression for the storage capacity for a given recall error rate

Error analysis of expressive analogy task in Spanish-English bilingual school age children with and without specific language impairment

textPurpose: The relational shift hypothesis (RSH) states that, as children age, the way in which they interpret analogies shifts from a focus on object similarities to relational aspects of objects. This study investigated the validity of the RSH by describing the error patterns of typically developing (TD), low normal (LN), and language impaired (LI) bilingual school-age children when completing an expressive analogy task in A:B::C:D format (e.g. good:bad::happy:_____) in English and Spanish. Method: Participants included a total of 49 Spanish-English bilingual children between the ages of 7;4 and 8; 9 (mean = 8; 1). Ten children were identified as LI, ten scored in the LN range, and 29 were TD. Children were administered English and Spanish versions of the item twice, initially during the second grade and once again approximately one year later. Responses were recorded verbatim and coded as correct (C), thematic/category error (THEM/CAT), wrong object, correct relationship error (WO-CR), unrelated error (UNREL), or repetition/no response (REP/NR). Results: A repeated measures ANOVA was used to compare children’s analogy scores by time, ability, and language. Results demonstrated significant differences for ability. Four chi square tests investigated the error patterns of TD, LN, and LI bilingual children in English and Spanish. We compared responses provided children by response type (C, THEM/CAT, WO-CR, UNREL, or REP/NR). Results from the Spanish analogical reasoning task indicated a decrease in THEM/CAT with age for the LN and TD children. Results from the English analogical reasoning task were inconsistent. Conclusions: Results provide partial support for the RSH in LN and TD children, but not in children with LI. This difference in error patterns may provide insight into the validity of the RSH in bilingual children with specific language impairment and typically developing second language learners.Communication Sciences and Disorder

Positional information, positional error, and read-out precision in morphogenesis: a mathematical framework

The concept of positional information is central to our understanding of how cells in a multicellular structure determine their developmental fates. Nevertheless, positional information has neither been defined mathematically nor quantified in a principled way. Here we provide an information-theoretic definition in the context of developmental gene expression patterns and examine which features of expression patterns increase or decrease positional information. We connect positional information with the concept of positional error and develop tools to directly measure information and error from experimental data. We illustrate our framework for the case of gap gene expression patterns in the early Drosophila embryo and show how information that is distributed among only four genes is sufficient to determine developmental fates with single cell resolution. Our approach can be generalized to a variety of different model systems; procedures and examples are discussed in detail

Order Statistics Based List Decoding Techniques for Linear Binary Block Codes

The order statistics based list decoding techniques for linear binary block codes of small to medium block length are investigated. The construction of the list of the test error patterns is considered. The original order statistics decoding is generalized by assuming segmentation of the most reliable independent positions of the received bits. The segmentation is shown to overcome several drawbacks of the original order statistics decoding. The complexity of the order statistics based decoding is further reduced by assuming a partial ordering of the received bits in order to avoid the complex Gauss elimination. The probability of the test error patterns in the decoding list is derived. The bit error rate performance and the decoding complexity trade-off of the proposed decoding algorithms is studied by computer simulations. Numerical examples show that, in some cases, the proposed decoding schemes are superior to the original order statistics decoding in terms of both the bit error rate performance as well as the decoding complexity.Comment: 17 pages, 2 tables, 6 figures, submitted to IEEE Transactions on Information Theor