Coding Theory Research Articles

Could Legitimation Code Theory offer practical insight for teaching disciplinary knowledge? A case study in geography

AbstractThis article explores the relationship between disciplinary knowledge and subject pedagogy, utilising Maton's Legitimation Code Theory (LCT). It suggests ways that LCT could help facilitate deeper communication both within and between subject communities, providing a conceptual framework that gets beneath empirical manifestations to identify epistemic and semantic principles that generate those modalities. LCT’s potential to represent systematised concepts diagrammatically may also enable wider communication. Following this, a small‐scale case study is presented exploring the nature of the knowledge in the core topic Changing Places in the new geography A‐Level (UK). This argued that what could be understood as a mismatch of expectations and dispositions by teachers and students can be understood more fundamentally as a contradiction (a ‘code clash’) regarding what counts as the ‘rules of the game’ within geography. The article ends by outlining implications for school geography rooted in the logic of the knowledge being pedagogically preinscribed, that is, that different knowledges’ distinct epistemologies have real implications for how that knowledge should be recontextualised, reproduced and evaluated. It concludes that the exam board's recontextualisation of this disciplinary knowledge is not merely a simpler interpretation; it is more significantly a different ideal ‘knower’ that is being nurtured and examined.

A survey on single server private information retrieval in a coding theory perspective

In this paper, we present a new perspective of single server private information retrieval (PIR) schemes by using the notion of linear error-correcting codes. Many of the known single server schemes are based on taking linear combinations between database elements and the query elements. Using the theory of linear codes, we develop a generic framework that formalizes all such PIR schemes. This generic framework provides an appropriate setup to analyze the security of such PIR schemes. In fact, we describe some known PIR schemes with respect to this code-based framework, and present the weaknesses of the broken PIR schemes in a unified point of view.

Predictive Coding Over the Lifespan: Increased Reliance on Perceptual Priors in Older Adults-A Magnetoencephalography and Dynamic Causal Modeling Study.

Aging is accompanied by unisensory decline. To compensate for this, two complementary strategies are potentially relied upon increasingly: first, older adults integrate more information from different sensory organs. Second, according to the predictive coding (PC) model, we form “templates” (internal models or “priors”) of the environment through our experiences. It is through increased life experience that older adults may rely more on these templates compared to younger adults. Multisensory integration and predictive coding would be effective strategies for the perception of near-threshold stimuli, which may however come at the cost of integrating irrelevant information. Both strategies can be studied in multisensory illusions because these require the integration of different sensory information, as well as an internal model of the world that can take precedence over sensory input. Here, we elicited a classic multisensory illusion, the sound-induced flash illusion, in younger (mean: 27 years, N = 25) and older (mean: 67 years, N = 28) adult participants while recording the magnetoencephalogram. Older adults perceived more illusions than younger adults. Older adults had increased pre-stimulus beta-band activity compared to younger adults as predicted by microcircuit theories of predictive coding, which suggest priors and predictions are linked to beta-band activity. Transfer entropy analysis and dynamic causal modeling of pre-stimulus magnetoencephalography data revealed a stronger illusion-related modulation of cross-modal connectivity from auditory to visual cortices in older compared to younger adults. We interpret this as the neural correlate of increased reliance on a cross-modal predictive template in older adults leading to the illusory percept.

Low-Delay FPGA-Based Implementation of Finite Field Multipliers

Arithmetic operations over binary extension fields GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) have many important applications in domains such as cryptography, code theory and digital signal processing. These applications must be fast, so low-delay implementations of arithmetic circuits are required. Among GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) arithmetic operations, field multiplication is considered the most important one. For hardware implementation of multiplication over binary finite fields, irreducible trinomials and pentanomials are normally used. In this brief, low-delay FPGA-based implementations of bit-parallel GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) polynomial basis multipliers are presented, where a new multiplier based on irreducible trinomials is given. Several post-place and route implementation results in Xilinx Artix-7 FPGA for different GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) finite fields are reported. Experimental results show that the proposed multiplier exhibits the best delay, with a delay improvement of up to 4.7%, and the second best Area×Time complexities when compared with similar multipliers found in the literature.

Structural and contextual priors affect visual search in children with and without autism.

Bayesian predictive coding theories of autism spectrum disorder propose that impaired acquisition or a broader shape of prior probability distributions lies at the core of the condition. However, we still know very little about how probability distributions are learned and encoded by children, let alone children with autism. Here, we take advantage of a recently developed distribution learning paradigm to characterize how children with and without autism acquire information about probability distributions. Twenty-four autistic and 25-matched neurotypical children searched for an odd-one-out target among a set of distractor lines with orientations sampled from a Gaussian distribution repeated across multiple trials to allow for learning of the parameters (mean and variance) of the distribution. We could measure the width (variance) of the participant's encoded distribution by introducing a target-distractor role-reversal while varying the similarity between target and previous distractor mean. Both groups performed similarly on the visual search task and learned the distractor distribution to a similar extent. However, the variance learned was much broader than the one presented, consistent with less informative priors in children irrespective of autism diagnosis. These findings have important implications for Bayesian accounts of perception throughout development, and Bayesian accounts of autism specifically. LAY SUMMARY: Recent theories about the underlying cognitive mechanisms of autism propose that the way autistic individuals estimate variability or uncertainty in their perceptual environment may differ from how typical individuals do so. Children had to search an oddly tilted line in a set of lines pointing in different directions, and based on their response times we examined how they learned about the variability in a set of objects. We found that autistic children learn variability as well as typical children, but both groups learn with less precision than typical adults do on the same task.

Single Mothers’ Tendency to Remarry and the Obstacles: A Case on Basic Theory

Background: Human beings are a social species. Marriage is an important step in human life because many of the main human needs are met after marriage. The purpose of this research was to study the tendency of single mothers to remarry, and the variables and barriers involved. The conceptual framework for this study involves a mix of social concept theories, Maslow’s necessity hierarchy, Eisen’s reasoning theory, and Lewin’s Psychological Field Theory on barriers. Methods: The current study used a qualitative technique (fundamental theory) for coding, editing, categorizing, and analyzing information. To the best of our knowledge, a fundamental theory analytical technique has been used to classify and evaluate information. For this purpose, eleven single mothers (all with kids) were surveyed in Gonabad city. Despite their inclination to build a connection between structure and interaction and by mixing information with theory, the scientists attempted to design the procedure for mothers’ celibacy. Results: The data analysis included coding of three main core levels, eleven first-category items, and 27-second category items. The main focus was feminism, social isolation, inappropriate cultural structure, social suicide, and the display of spiritual and material life. Conclusion: The results showed that single mothers not only tend to remarry but also seriously need to be married. However, the inner and outer variables stop them from remarrying. Therefore, they are looking for a replacement that leads to their social isolation. Facilitative strategies to pave the way for their remarriage can therefore assist them considerably.

“ТРАМПІЗМИ” – ВІД ОКАЗІОНАЛІЗМУ ДО УЗУАЛЬНОЇ ЛЕКСИЧНОЇ ОДИНИЦІ

The article deals with the problem of occasional lexical units used by the 45th American President D.Trump in his political speeches, and their further possible progress to the status of usual words generally used in the American variant of the English language. Political discourse is taken as an object of the research due to its popularity, and the highest circulation in the modern democratic society that, with the help of its characteristics, guarantees maximum promulgation of the occasional words. It is stressed in the article that the study cannot be purely done in linguistic terms, as the very progress from occasional to the usual words covers the aspects of pedagogics, psychology, neuropsychology, neurodidactics. To trespass the borderline between occasional and usual, it is surely badly needed to be repeated scores of times – the task that is fully performed by the political and media discourses in concatenation - but still it alone is not enough. The newly created word should correspond not only to the communicative demands of a speaker-creator, but also with the views and beliefs of the audience. Thus, it catches declarative, procedural, semantic, and episodic / emotional memory of the addressee. Altogether, it should be of some significance as in the context of the information given by the politician, so from the prospective of the very addressee (his life values and experience). Dual Coding Theory involving the activity of verbal and non-verbal systems comes to the fore to explain the principles of successful memorization of occasional lexical units coming to life through political discourse.

Trips, tours and random walks: using Legitimation Code Theory to understand spontaneous teachable moments

Trips, tours and random walks: using Legitimation Code Theory to understand spontaneous teachable moments

A Study on the Application of Code Theory in the Decorative Design of Taiwan Bamboo Tube Furniture

Taiwan has an abundance of the environmentally-friendly resource bamboo, as well as long-standing Chinese bamboo customs and culture. As a result, modest bamboo furniture accompanies the simple life of Taiwan’s residents. However, as people prosper, consumers are beginning to choose bamboo furniture for its decorative design and style more than for its practicality. Therefore, the symbolic application of decorative art is worthy of research and discussion. The purpose of this research is to use codes to explore the decorative design of bamboo tube furniture and enlighten Taiwan’s new direction of bamboo furniture design in the future. Through literature content analysis, field expert interviews, and practical experience participation methods, in addition to using function codes, emotion codes, transmission codes, and identification codes to interpret the Han Dynasty’s Liushu and Chinese bamboo cultural content, we verify the relevance of the code and bamboo furniture decoration and provide interactive new thinking and creative products.

AI Uncertainty Based on Rademacher Complexity and Shannon Entropy

In this paper from communication channel coding perspective we are able to present both a theoretical and practical discussion of AI’s uncertainty, capacity and evolution for pattern classification based on the classical Rademacher complexity and Shannon entropy. First AI capacity is defined as in communication channels. It is shown qualitatively that the classical Rademacher complexity and Shannon rate in communication theory is closely related by their definitions. Secondly based on the Shannon mathematical theory on communication coding, we derive several sufficient and necessary conditions for an AI’s error rate approaching zero in classifications problems. A 1/2 criteria on Shannon entropy is derived in this paper so that error rate can approach zero or is zero for AI pattern classification problems. Last but not least, we show our analysis and theory by providing examples of AI pattern classifications with error rate approaching zero or being zero. Impact Statement: Error rate control of AI pattern classification is crucial in many lives related AI applications. AI uncertainty, capacity and evolution are investigated in this paper. Sufficient/necessary conditions for AI’s error rate approaching zero are derived based on Shannon’s communication coding theory. Zero error rate and zero error rate approaching AI design methodology for pattern classifications are illustrated using Shannon’s coding theory. Our method shows how to control the error rate of AI, how to measure the capacity of AI and how to evolve AI into higher levels. Index Terms: Rademacher Complexity, Shannon Theory, Shannon Entropy, Vapnik-Cheronenkis (VC) dimension.

The encoding of stochastic regularities is facilitated by action-effect predictions

Our brains continuously build and update predictive models of the world, sources of prediction being drawn for example from sensory regularities and/or our own actions. Yet, recent results in the auditory system indicate that stochastic regularities may not be easily encoded when a rare medium pitch deviant is presented between frequent high and low pitch standard sounds in random order, as reflected in the lack of sensory prediction error event-related potentials [i.e., mismatch negativity (MMN)]. We wanted to test the implication of the predictive coding theory that predictions based on higher-order generative models—here, based on action intention, are fed top-down in the hierarchy to sensory levels. Participants produced random sequences of high and low pitch sounds by button presses in two conditions: In a “specific” condition, one button produced high and the other low pitch sounds; in an “unspecific” condition, both buttons randomly produced high or low-pitch sounds. Rare medium pitch deviants elicited larger MMN and N2 responses in the “specific” compared to the “unspecific” condition, despite equal sound probabilities. These results thus demonstrate that action-effect predictions can boost stochastic regularity-based predictions and engage higher-order deviance detection processes, extending previous notions on the role of action predictions at sensory levels.

Some direction: towards a C21 secondary school curriculum

ABSTRACT In this third and final paper from the Delphi study One Direction, we report on participants’ responses to four secondary school music curriculum scenarios. These scenarios present four possible directions for a C21 secondary school music curriculum. The scenarios were devised from a combination of ideas derived from the data from the earlier stages of the study (McPhail, G., and J. McNeill. 2019. “One Direction: A Future for Secondary School Music Education?” Music Education Research 21 (4): 359–370; McNeill, J., and G. McPhail. 2020. “One Direction: Strategic challenges for Twenty-first Century Secondary School Music.” Music Education Research 22 (4): 432–446) and the concept of specialisation from Maton’s Legitimation Code Theory. By asking an international panel of leading music education researchers and teachers to respond to the scenarios, we are able to argue that ‘one direction’ is unlikely to emerge for secondary school music education, but we discuss the responses, and the scenario dimensions regarded as most likely and desirable. What appears certain is that there will be a continuing weakening of the boundaries between types of knowledge and stylistic arenas suggesting a dialectic relationship between the legitimating principles most valued.

Infinite combinatorics in mathematical biology

Is it possible to apply infinite combinatorics and (infinite) set theory in theoretical biology? We do not know the answer yet but in this article we try to present some techniques from infinite combinatorics and set theory that have been used over the last decades in order to prove existence results and independence theorems in algebra and that might have the flexibility and generality to be also used in theoretical biology. In particular, we will introduce the theory of forcing and an algebraic construction technique based on trees and forests using infinite binary sequences. We will also present an overview of the theory of circular codes. Such codes had been found in the genetic information and are assumed to play an important role in error detecting and error correcting mechanisms during the process of translation. Finally, examples and constructions of infinite mixed circular codes using binary sequences hopefully show some similarity between these theories - a starting point for future applications.

Matroidal Entropy Functions: A Quartet of Theories of Information, Matroid, Design, and Coding.

In this paper, we study the entropy functions on extreme rays of the polymatroidal region which contain a matroid, i.e., matroidal entropy functions. We introduce variable strength orthogonal arrays indexed by a connected matroid M and positive integer v which can be regarded as expanding the classic combinatorial structure orthogonal arrays. It is interesting that they are equivalent to the partition-representations of the matroid M with degree v and the almost affine codes. Thus, a synergy among four fields, i.e., information theory, matroid theory, combinatorial design, and coding theory is developed, which may lead to potential applications in information problems such as network coding and secret-sharing. Leveraging the construction of variable strength orthogonal arrays, we characterize all matroidal entropy functions of order with the exception of and for some v.

Gaussian sums, hyper Eisenstein sums and Jacobi sums over a local ring and their applications

It is well known that any finite commutative ring is isomorphic to a direct product of local rings via the Chinese remainder theorem. Hence, there is a great significance to the study of character sums over local rings. Character sums over finite rings have applications that are analogous to the applications of character sums over finite fields. In particular, character sums over local rings have many applications in algebraic coding theory. In this paper, we firstly present an explicit description on additive characters and multiplicative characters over a certain local ring. Then we study Gaussian sums, hyper Eisenstein sums and Jacobi sums over a certain local ring and explore their properties. It is worth mentioning that we are the first to define Eisenstein sums and Jacobi sums over this local ring. Moreover, we present a connection between hyper Eisenstein sums over this local ring and Gaussian sums over finite fields, which allows us to give the absolute value of hyper Eisenstein sums over this local ring. As an application, several classes of codebooks with new parameters are presented.

Limited Evidence for Sensory Prediction Error Responses in Visual Cortex of Macaques and Humans.

A recent formulation of predictive coding theory proposes that a subset of neurons in each cortical area encodes sensory prediction errors, the difference between predictions relayed from higher cortex and the sensory input. Here, we test for evidence of prediction error responses in spiking responses and local field potentials (LFP) recorded in primary visual cortex and area V4 of macaque monkeys, and in complementary electroencephalographic (EEG) scalp recordings in human participants. We presented a fixed sequence of visual stimuli on most trials, and violated the expected ordering on a small subset of trials. Under predictive coding theory, pattern-violating stimuli should trigger robust prediction errors, but we found that spiking, LFP and EEG responses to expected and pattern-violating stimuli were nearly identical. Our results challenge the assertion that a fundamental computational motif in sensory cortex is to signal prediction errors, at least those based on predictions derived from temporal patterns of visual stimulation.

Character sums over a non-chain ring and their applications

Some valuable results over rings have a promising utilization in coding theory and error-correcting code theory. In this paper, we study character sums over a certain non-chain ring and their applications in codebooks. There are two major ingredients in this study. The first ingredient is to investigate Gaussian sums, hyper Eisenstein sums, Jacobi sums over a certain non-chain ring and study the properties of these character sums. For their applications, the second ingredient is to present three classes of asymptotically optimal codebooks with respect to the Welch bound and a family of optimal codebooks with respect to the Levenshtein bound, which are constructed from character sums over a certain non-chain ring.

Codes in Rosenbloom-Tsfasman metric: A Survey

This paper gives a systematic survey of research carried out in the theory of codes equipped with Rosenbloom-Tsfasman metric. In classical coding theory setting, codes are investigated with respect to the Hamming metric which can efficiently address the communication problems arising from channels in which channel noise generates equiprobable errors. But however, not all the real world channels are of that nature, especially, when the possible errors form patterns of a specific shape. Rosenbloom and Tsfasman introduced a non-Hamming metric, called Rosenbloom-Tsfasman metric (RT-metric, in short) that can address the problem of reliable information transmission over parallel noisy channels. Martin, Stinson and Skriganov independently introduced the same metric in the context of the theory of uniform distributions. As this metric happened to be a generalization of the classical Hamming metric, it has attracted so much attention from the coding theory research community and as a result a lot of work has been done in this line of research over the past 3 decades. In this paper we would like to present the key developments in the field of codes with RT-metric.

Quantum stabilizer codes, lattices, and CFTs

There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E8 theory, which is based on the root lattice of E8 understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples.

Modulatory Effects of Prediction Accuracy on Electroencephalographic Brain Activity During Prediction.

Prediction is essential for the efficiency of many cognitive processes; however, this process is not always perfect. Predictive coding theory suggests that the brain generates and updates a prediction to respond to an upcoming event. Although an electrophysiological index of prediction, the stimulus preceding negativity (SPN), has been reported, it remains unknown whether the SPN reflects the prediction accuracy, or whether it is associated with the prediction error, which corresponds to a mismatch between a prediction and an actual input. Thus, the present study aimed to investigate this question using electroencephalography (EEG). Participants were asked to predict the original pictures from pictures that had undergone different levels of pixelation. The SPN amplitude was affected by the level of pixelation and correlated with the subjective evaluation of the prediction accuracy. Furthermore, late positive components (LPC) were negatively correlated with SPN. These results suggest that the amplitude of SPN reflects the prediction accuracy; more accurate prediction increases the SPN and reduces the prediction error, resulting in reduced LPC amplitudes.