Cortical processing of musical pitch as reflected by behavioural and electrophysiological evidence

In a musical context, the pitch of sounds is encoded according to domain-general principles not confined to music or even to audition overall but common to other perceptual and cognitive processes (such as multiple pattern encoding and feature integration), and to domain-specific and culture-specific properties related to a particular musical system only (such as the pitch steps of the Western tonal system). The studies included in this thesis shed light on the processing stages during which pitch encoding occurs on the basis of both domain-general and music-specific properties, and elucidate the putative brain mechanisms underlying pitch-related music perception. Study I showed, in subjects without formal musical education, that the pitch and timbre of multiple sounds are integrated as unified object representations in sensory memory before attentional intervention. Similarly, multiple pattern pitches are simultaneously maintained in non-musicians' sensory memory (Study II). These findings demonstrate the degree of sophistication of pitch processing at the sensory memory stage, requiring neither attention nor any special expertise of the subjects. Furthermore, music- and culture-specific properties, such as the pitch steps of the equal-tempered musical scale, are automatically discriminated in sensory memory even by subjects without formal musical education (Studies III and IV). The cognitive processing of pitch according to culture-specific musical-scale schemata hence occurs as early as at the sensory-memory stage of pitch analysis. Exposure and cortical plasticity seem to be involved in musical pitch encoding. For instance, after only one hour of laboratory training, the neural representations of pitch in the auditory cortex are altered (Study V). However, faulty brain mechanisms for attentive processing of fine-grained pitch steps lead to inborn deficits in music perception and recognition such as those encountered in congenital amusia (Study VI). These findings suggest that predispositions for exact pitch-step discrimination together with long-term exposure to music govern the acquisition of the automatized schematic knowledge of the music of a particular culture that even non-musicians possess.

Scheduling a Burn-In Oven with Non-Agreeable Release Times and Due Dates

We consider the problem of scheduling semiconductor burn-in operations, where burn-in ovens are modelled as batch processing machines. Most of the studies assume that ready times and due dates of jobs are agreeable (i.e., ri < rj implies di ≤ dj). In many real world applications, the agreeable property assumption does not hold. Therefore, in this paper, scheduling of a single burn-in oven with non-agreeable release times and due dates along with non-identical job sizes as well as non-identical processing of time problem is formulated as a Non-Linear (0-1) Integer Programming optimisation problem. The objective measure of the problem is minimising the maximum completion time (makespan) of all jobs. Due to computational intractability, we have proposed four variants of a two-phase greedy heuristic algorithm. Computational experiments indicate that two out of four proposed algorithms have excellent average performance and also capable of solving any large-scale real life problems with a relatively low computational effort on a Pentium IV computer.

从字形到语义：跨语言Stroop效应的ERP研究

According to the influential dual-route model of reading (Coltheart, Rastle et al. 2001), there are two routes to access the meaning of visual words: one directly by orthography (orthography-semantic) and the other indirectly via the phonology (phonology-semantic). Because of the dramatic difference between written Chinese and alphabetical languages, it is still on debate whether Chinese readers have the same semantic activation processes as readers of alphabetical languages. In this study, the semantic activation processes in alphabetical German and logographic Chinese were compared. Since the N450 for incongruent color words in the Stroop tasks was induced by the semantic conflict between the meaning of the incongruent color words and color naming, this component could be taken as an index for semantic activation of incongruent color words in Stroop tasks. Two cross-script Stroop experiments were adopted to investigate the semantic activation processes in Chinese and German. The first experiment focused on the the role of phonology, while the second one focused on the realative importance of orthography. Cultural differences in cognitive processing between individuals in western and eastern countries have been found (Nisbett & Miyamoto, 2005). In order to exclude potential differences in basic cognitive processes like visual discrimination capabilities during reading, a visual Oddball experiment with non-lexical materials was conducted with all participants. However, as indicated by the P300 elicited by deviant stimuli in both groups, no group difference was observed. In the first Stroop experiments, color words (e.g., “green”), color-word associates (e.g., “grass”), and homophones of color words were used. These words were embedded into color patches with either congruent color (e.g. word “green” in green color patch) or incongruent colors (e.g. word “green” in either red or yellow or blue color patch). The key point is to observe whether homophones in both languages could induce similar behavioral and ERP Stroop effects to that induced by color words. It was also interesting to observe to which extent the N450 was related to the semantic conflicts. Nineteen Chinese adult readers and twenty German adult readers were asked to respond to the back color of these words in the Stroop experiment in their native languages by pressing the corresponding keys. In the behavioral data, incongruent conditions (incongruent color words, incongruent color-word associates, incongruent homophones) had significantly longer reaction times as compared to corresponding congruent conditions. All incongruent conditions in the Geman group elicited an N450 in the 400 to 500 ms time window. In the Chinese group, the N450 in the same time window was also observed for the incongruent color words and incongruent color-word associates. These results indicated that the N450 was very sensitive to semantic conflict-even words with semantic association to colors (e.g. “grass”) could elicite similar N450. However, the N450 was absent for incongruent homophones of color words in the Chinese group. Instead, in a later time window (600-800 ms), incongruent homophones elicited a positivity over left posterior regions as compared to congruent homophones. Similar positivity was also observed for color words in the 700 to 1000 ms time window in the Chinese group and 600 to 1000 ms time window for incongruent color words and homophones in the Geman group. These results indicate that phonology plays an important role in Geman semantic activation processes, but not in Chinese. In the second Stroop experiment, color words and pseudowords which had similiar visual shape to color words in both languages were used as materials. Another group of eighteen Chinese and twenty Germans were involved in the Stroop experiment in their native languages.The ERPs were recorded during their performance. In the behavioral data, strong and comparable Stroop effects (as counted by substract the reaction times in the congruent conditions from reaction times in the incongruent conditions) were observed. In the ERP data, both incongruent color words and incongruent pseudowords elicited an N450 over the whole brain scalp in both groups. These results indicated that orthography played an equally important role in semantic activation processes in both languages. The results of the two Stroop experiments support the view that the semantic activation process in Chiense readers differs significantly from that in German readers. The former rely mainly on the direct route (orthography-semantic), while the latter use both direct route and incirect route (phonology-semantic). These findings also indicate that the characteritics of different languages shape the semantic activation processes.

Laminar Cortical Dynamics of Cognitive and Motor Working Memory, Sequence Learning and Performance: Toward a Unified Theory of How the Cerebral Cortex Works

How do the layered circuits of prefrontal and motor cortex carry out working memory storage, sequence learning, and voluntary sequential item selection and performance? A neural model called LIST PARSE is presented to explain and quantitatively simulate cognitive data about both immediate serial recall and free recall, including bowing of the serial position performance curves, error-type distributions, temporal limitations upon recall, and list length effects. The model also qualitatively explains cognitive effects related to attentional modulation, temporal grouping, variable presentation rates, phonemic similarity, presentation of non-words, word frequency/item familiarity and list strength, distracters and modality effects. In addition, the model quantitatively simulates neurophysiological data from the macaque prefrontal cortex obtained during sequential sensory-motor imitation and planned performance. The article further develops a theory concerning how the cerebral cortex works by showing how variations of the laminar circuits that have previously clarified how the visual cortex sees can also support cognitive processing of sequentially organized behaviors.

Improvement in visual search with practice: mapping learning-related changes in neurocognitive stages of processing.

Practice can improve performance on visual search tasks; the neural mechanisms underlying such improvements, however, are not clear. Response time typically shortens with practice, but which components of the stimulus-response processing chain facilitate this behavioral change? Improved search performance could result from enhancements in various cognitive processing stages, including (1) sensory processing, (2) attentional allocation, (3) target discrimination, (4) motor-response preparation, and/or (5) response execution. We measured event-related potentials (ERPs) as human participants completed a five-day visual-search protocol in which they reported the orientation of a color popout target within an array of ellipses. We assessed changes in behavioral performance and in ERP components associated with various stages of processing. After practice, response time decreased in all participants (while accuracy remained consistent), and electrophysiological measures revealed modulation of several ERP components. First, amplitudes of the early sensory-evoked N1 component at 150 ms increased bilaterally, indicating enhanced visual sensory processing of the array. Second, the negative-polarity posterior-contralateral component (N2pc, 170-250 ms) was earlier and larger, demonstrating enhanced attentional orienting. Third, the amplitude of the sustained posterior contralateral negativity component (SPCN, 300-400 ms) decreased, indicating facilitated target discrimination. Finally, faster motor-response preparation and execution were observed after practice, as indicated by latency changes in both the stimulus-locked and response-locked lateralized readiness potentials (LRPs). These electrophysiological results delineate the functional plasticity in key mechanisms underlying visual search with high temporal resolution and illustrate how practice influences various cognitive and neural processing stages leading to enhanced behavioral performance.

Rise time and formant transition duration in the discrimination of speech sounds: The Ba–Wa distinction in developmental dyslexia

Across languages, children with developmental dyslexia have a specific difficulty with the neural representation of the sound structure (phonological structure) of speech. One likely cause of their difficulties with phonology is a perceptual difficulty in auditory temporal processing (Tallal, 1980). Tallal (1980) proposed that basic auditory processing of brief, rapidly successive acoustic changes is compromised in dyslexia, thereby affecting phonetic discrimination (e.g. discriminating /b/ from /d/) via impaired discrimination of formant transitions (rapid acoustic changes in frequency and intensity). However, an alternative auditory temporal hypothesis is that the basic auditory processing of the slower amplitude modulation cues in speech is compromised (Goswami , 2002). Here, we contrast children's perception of a synthetic speech contrast (ba/wa) when it is based on the speed of the rate of change of frequency information (formant transition duration) versus the speed of the rate of change of amplitude modulation (rise time). We show that children with dyslexia have excellent phonetic discrimination based on formant transition duration, but poor phonetic discrimination based on envelope cues. The results explain why phonetic discrimination may be allophonic in developmental dyslexia (Serniclaes , 2004), and suggest new avenues for the remediation of developmental dyslexia. © 2010 Blackwell Publishing Ltd.

Autonomic prediction of error-related ERP components in perceptual and conceptual tasks in older and younger adults

In studies of cognitive processing, the allocation of attention has been consistently linked to subtle, phasic adjustments in autonomic control. Both autonomic control of heart rate and control of the allocation of attention are known to decline with age. It is not known, however, whether characteristic individual differences in autonomic control and the ability to control attention are closely linked. To test this, a measure of parasympathetic function, vagal tone (VT) was computed from cardiac recordings from older and younger adults taken before and during performance of two attentiondemanding tasks - the Eriksen visual flanker task and the source memory task. Both tasks elicited event-related potentials (ERPs) that accompany errors, i.e., error-related negativities (ERNs) and error positivities (Pe's). The ERN is a negative deflection in the ERP signal, time-locked to responses made on incorrect trials, likely generated in the anterior cingulate. It is followed immediately by the Pe, a broad, positive deflection which may reflect conscious awareness of having committed an error. Age-attenuation ofERN amplitude has previously been found in paradigms with simple stimulus-response mappings, such as the flanker task, but has rarely been examined in more complex, conceptual tasks. Until now, there have been no reports of its being investigated in a source monitoring task. Age-attenuation of the ERN component was observed in both tasks. Results also indicated that the ERNs generated in these two tasks were generally comparable for young adults. For older adults, however, the ERN from the source monitoring task was not only shallower, but incorporated more frontal processing, apparently reflecting task demands. The error positivities elicited by 3 the two tasks were not comparable, however, and age-attenuation of the Pe was seen only in the more perceptual flanker task. For younger adults, it was Pe scalp topography that seemed to reflect task demands, being maximal over central parietal areas in the flanker task, but over very frontal areas in the source monitoring task. With respect to vagal tone, in the flanker task, neither the number of errors nor ERP amplitudes were predicted by baseline or on-task vagal tone measures. However, in the more difficult source memory task, lower VT was marginally associated with greater numbers of source memory errors in the older group. Thus, for older adults, relatively low levels of parasympathetic control over cardiac response coincided with poorer source memory discrimination. In both groups, lower levels of baseline VT were associated with larger amplitude ERNs, and smaller amplitude Pe's. Thus, low VT was associated in a conceptual task with a greater "emergency response" to errors, and at the same time, reduced awareness of having made them. The efficiency of an individual's complex cognitive processing was therefore associated with the flexibility of parasympathetic control of heart rate, in response to a cognitively challenging task.

The effects of postural threat on cognitive strategies used to maintain upright stance

It is well established that postural threat modifies postural control, although little is known regarding the underlying mechanism(s) responsible. It is possible that changes in postural control under conditions of elevated postural threat result from alterations in cognitive strategies. The purpose of this study was to determine the influence of elevated postural threat on cognitive strategies and to determine the relationship between postural control, psychological, and cognitive measures. It was hypothesized that elevated postural threat would cause a shift to more conscious control of posture. It was also expected that a relationship between fear of falling and postural control would exist that could be explained by changes in conscious control of posture. Forty-eight healthy young adults stood on a force plate at two different surface heights: ground level (LOW) and 3.2m above ground level (HIGH). Center of pressure (COP) summary measures calculated to quantify postural control were the mean position (AP-COP MP), root mean square (AP-COP RMS) and mean power frequency (AP-COP MPF) in the anteriorposterior direction. Trunk sway measures calculated in the pitch direction were trunk angle and trunk velocity. Psychological measures including perceived balance confidence, perceived fear of falling, perceived anxiety, and perceived stability were self reported. As a physiological indicator of anxiety, electrodermal activity was collected. The cognitive strategies assessed were movement reinvestment and attention focus. A modified state-sp-ecific version of the Movement Specific Reinvestment Scale was used to measure conscious motor processing (CMP) and movement self-consciousness (MSC). An attention focus questionnaire was developed to assess the amount of attention directed to internal and external sources. An effect of postural threat on cognitive strategies was observed as participants reported more conscious control and a greater concern or worry about their posture at the HIGH postural threat condition as well as an increased internal and external focus of attention. In addition changes in postural control, psychological, and physiological measures were found. The participants leaned away from the edge of the platform, the frequency of their postural adjustments increased, and the velocity of their trunk movements increased. Participants felt less confident, more fearful, more anxious, and less stable with an accompanying increase in physiological anxiety. Significant correlations between perceived anxiety, AP-COP MP, and cognitive measures revealed a possible relationship that could be mediated by cognitive measures. It was found that with greater conscious motor processing, more movement self-consciousness, and a greater amount of attention focused externally there was a larger shift of the mean position away from the edge of the platform. This thesis provides evidence that postural threat can influence cognitive strategies causing a shift to more conscious control of movement which is associated with leaning away from the edge of the platform. Shifting the position of the body away from the direction of the postural threat may reflect a cognitive strategy to ensure safety in this situation due to the inability to employ a stepping strategy when standing on an elevated platform.

An Electrophysiological Investigation into the Role of Cognitive Control in the Attentional Blink

Accuracy at reporting a second-target (T2) is reduced if it is presented within approximately 500 ms of the first target (T1) – an attentional blink (AB). Early models explained the AB in terms of attentional limitations creating a processing bottleneck such that T2 processing would be impaired while T1 processing was ongoing. Theoretical models of the AB have more recently been expanded to include the role of cognitive control. In this dissertation I propose that cognitive control, defined as the optimization of information processing in order to achieve goals, is maladapted to the dual-task conditions of the AB task in that cognitive control optimizes the T1 goal, due to its temporal proximity, at the cost of T2. I start with the concept that the role of cognitive control is to serve goals, and that how goals are conceived of and the degree of motivation associated with those goals will determine whether cognitive control will create the condition that cause the AB. This leads to the hypothesis that electrophysiological measures of cognitive control and the degree of attentional investment resulting from cognitive control modulate the AB and explain individual differences in the AB. In a series of four studies feedback-related N2 amplitude, (reflecting individual differences in the strength of cognitive control), and event-related and resting alpha frequency oscillatory activity (reflecting degree of attentional investment), are used to explain both intra- and inter-individual variability in performance on the AB task. Results supported the hypothesis that stronger cognitive control and greater attentional investment are associated with larger AB magnitudes. Attentional investment, as measured by alpha frequency oscillations, and cognitive control, as measured by the feedback-related N2, did not relate to each other as hypothesized. It is proposed that instead of a measure of attentional investment alone, alpha frequency oscillatory activity actually reflects control over information processing over time, in other words the timing of attention. With this conceptualization, various aspects of cognitive control, either related to the management of goals (feedback-related N2) or the management of attention over time to meet goals, explain variability in the AB.

Prefrontal inhibition of threat processing reduces working memory interference

Bottom-up processes can interrupt ongoing cognitive processing in order to adaptively respond to emotional stimuli of high potential significance, such as those that threaten wellbeing. However it is vital that this interference can be modulated in certain contexts to focus on current tasks. Deficits in the ability to maintain the appropriate balance between cognitive and emotional demands can severely impact on day-to-day activities. This fMRI study examined this interaction between threat processing and cognition; 18 adult participants performed a visuospatial working memory (WM) task with two load conditions, in the presence and absence of anxiety induction by threat of electric shock. Threat of shock interfered with performance in the low cognitive load condition; however interference was eradicated under high load, consistent with engagement of emotion regulation mechanisms. Under low load the amygdala showed significant activation to threat of shock that was modulated by high cognitive load. A directed top-down control contrast identified two regions associated with top-down control; ventrolateral PFC and dorsal ACC. Dynamic causal modeling provided further evidence that under high cognitive load, top-down inhibition is exerted on the amygdala and its outputs to prefrontal regions. Additionally, we hypothesized that individual differences in a separate, non-emotional top-down control task would predict the recruitment of dorsal ACC and ventrolateral PFC during top-down control of threat. Consistent with this, performance on a separate dichotic listening task predicted dorsal ACC and ventrolateral PFC activation during high WM load under threat of shock, though activation in these regions did not directly correlate with WM performance. Together, the findings suggest that under high cognitive load and threat, top-down control is exerted by dACC and vlPFC to inhibit threat processing, thus enabling WM performance without threat-related interference.

Children's recall of words spoken in their first and second language : Effects of Signal-to-Noise Ratio and Reverberation Time

Speech perception runs smoothly and automatically when there is silence in the background, but when the speech signal is degraded by background noise or by reverberation, effortful cognitive processing is needed to compensate for the signal distortion. Previous research has typically investigated the effects of signal-to-noise ratio (SNR) and reverberation time in isolation, whilst few have looked at their interaction. In this study, we probed how reverberation time and SNR influence recall of words presented in participants' first- (L1) and second-language (L2). A total of 72 children (10 years old) participated in this study. The to-be-recalled wordlists were played back with two different reverberation times (0.3 and 1.2 s) crossed with two different SNRs (+3 dBA and +12 dBA). Children recalled fewer words when the spoken words were presented in L2 in comparison with recall of spoken words presented in L1. Words that were presented with a high SNR (+12 dBA) improved recall compared to a low SNR (+3 dBA). Reverberation time interacted with SNR to the effect that at +12 dB the shorter reverberation time improved recall, but at +3 dB it impaired recall. The effects of the physical sound variables (SNR and reverberation time) did not interact with language. © 2016 Hurtig, Keus van de Poll, Pekkola, Hygge, Ljung and Sörqvist.

Women's cognitive and emotional processing during sexual arousal : the effects of erotic film and absorption

Sexual arousal is a complex and dynamic element of women’s sexuality. Accounts vary, however most multidimensional models highlight the cognitive, affective and physiological components of the female sexual arousal response. While literature examining the peripheral physiological changes that occur during female sexual arousal abounds, there has been a dearth of literature pertaining to the cognitive and affective components. Thus, while many authors have included cognitive and emotions as independent components of the female sexual arousal response, there has been little empirical research to validate this approach. The aim of the current thesis was to examine the cognitive and affective components of female sexual arousal in more depth, investigating the nature of the relationship between these components under various experimental conditions. In order to do so, two integrated studies were conducted, each highlighting the effects of either external or internal variables on women’s subjective sexual arousal, absorption, positive affect and negative affect. Study One was designed to evaluate women’s emotional and cognitive processing of two elements of erotic film – foreplay and erotic context. 60 women were asked to report their subjective sexual arousal, absorption, positive affect and negative affect in response to one of four erotic film excerpts. The erotic excerpts varied in accordance with the degree of foreplay (low vs. high) depicted and the context in which the sexual activities took place (novel vs. habitual context). Women in the study responded more favourably to the high foreplay erotic film excerpt, subsequently reporting higher degrees of subjective sexual arousal, absorption and positive affect. Women also responded favourably to the erotic excerpt filmed in a novel context, reporting greater subjective sexual arousal as a result. The environment in which the sexual encounters were filmed failed to have an effect however, on women’s absorption or their positive or negative affect. The results of Study One suggest that stimulus specific variables, such as the degree of foreplay depicted, have a significant influence on female cognitive and emotional processing of erotic film. The results also suggest that a relationship exists between absorption, subjective sexual arousal and positive affect, albeit a correlational one. Specifically, there was evidence of parallel processing during sexual arousal, as participant reported sexual arousal, absorption and positive affect all increased and decreased in unison. Based on the results it was suggested that future research attempt to experimentally manipulate one of these variables, to examine its direct effect on the remaining variable. Thus, Study Two aimed to examine the effects of absorption on women’s cognitive and emotional processing of erotica. Study Two manipulated absorption at two levels (high vs. low), examining the impact of these states on participants’ subsequent absorption, subjective sexual arousal and positive and negative affect. 62 women were asked to read one of two sets of test session instructions. The first, participant-oriented instruction set, instructed participants to immerse themselves in the erotic film excerpt, as if they were active participants in the sexual exchange. The second, spectator-oriented instruction set, directed participants to observe and evaluate the erotic film. These instructions were designed to elicit high and low degrees of absorption, respectively. The utility of this approach when manipulating female absorption, was demonstrated by self reported ratings of absorption, given at the conclusion of the film presentation. Participants were also asked to report their subjective sexual arousal and positive and negative affect at the conclusion of the erotic film presentation. The findings of this study suggest that the adoption of a participant-oriented (high absorption) perspective elicits more favourable responses from participants than a spectator-oriented (low absorption) perspective, with participants in the former experimental group reporting greater degrees of subjective sexual arousal and positive affect. Negative affect was equivalent across experimental conditions, with the participants reporting that they experienced little to no aversive feelings during either of the experimental conditions. The results suggest that the degree to which a women immerses and absorbs herself in a sexual stimulus has a significant impact in her subsequent cognitive and affective processing of that stimulus. More specifically, it appears that women respond more favourably when they are highly absorbed and immersed in a stimulus, reporting greater subjective sexual arousal and positive affect. Overall, the results of Studies One and Two highlight the dynamic and complex nature of female sexual arousal. It appears that women have definite cognitive and affective responses to sexual stimuli. The magnitude of these responses may be mediated by a number of factors however, including the intrinsic qualities of the stimulus and the degree to which the woman attends to the stimulus. Both these variables act to either enhance or inhibit the sexual arousal response. There results have important implications for current sexuality literature. While women’s cognitions and emotions in response to erotic film were generally highly correlated, in some instances they differed, warranting their inclusion as separate elements in models of female sexual arousal. Furthermore, it might be suggested that the inclusion of an additional variable – absorption – into current models of female sexual arousal would prove beneficial, aiding researchers to better understand and predict the arousal process. As such, recommendations are made for a revised model of female sexual arousal. In terms of future directions, the results of the present thesis have implications for the treatment of sexual dysfunctions, suggesting that clinicians need to understand the internal and external variables that might contribute to the aetiology and maintenance of their presenting problems.

Mild cognitive impairment (part 1): Clinical characteristics and predictors of dementia

Objective: To critically review and evaluate existing knowledge on the conceptual limits and clinical usefulness of the diagnosis of mild cognitive impairment (MCI) and the neuropsychological assessment and short- and long-term prognosis thereof. Methods: We conducted a systematic search of the PubMed and Web of Science electronic databases, limited to articles published in English between 1999 and 2012. Based on the search terms mild cognitive impairment or MCI and epidemiology or diagnosis, we retrieved 1,698 articles, of which 248 were critically eligible (cross-sectional and longitudinal studies); the abstracts of the remaining 1,450 articles were also reviewed. Results: A critical review on the MCI construct is provided, including conceptual and diagnostic aspects; epidemiological relevance; clinical assessment; prognosis; and outcome. The distinct definitions of cognitive impairment, MCI included, yield clinically heterogeneous groups of individuals. Those who will eventually progress to dementia may present with symptoms consistent with the definition of MCI; conversely, individuals with MCI may remain stable or return to normal cognitive function. Conclusion: On clinical grounds, the cross-sectional diagnosis of MCI has limited prognostic relevance. The characterization of persistent and/or progressive cognitive deficits over time is a better approach for identification of cases at the pre-dementia stages, particularly if these cognitive abnormalities are consistent with the natural history of incipient Alzheimer's disease. © 2013 Associação Brasileira de Psiquiatria.

Construction of perfect space-time codes

In this work, we propose an innovative methodology to extend the construction of minimum and non-minimum delay perfect codes as a subset of cyclic division algebras over ℚ(ζ3), where the signal constellations are isomorphic to the hexagonal An 2 -rotated lattice, for any channel of any dimension n such that gcd{n, 3) = 1.