Maximum Weight Spectrum Codes

In the recent work \cite{shi18}, a combinatorial problem concerning linear codes over a finite field \F_q was introduced. In that work the authors studied the weight set of an $[n,k]_q$ linear code, that is the set of non-zero distinct Hamming weights, showing that its cardinality is upper bounded by $\frac{q^k-1}{q-1}$. They showed that this bound was sharp in the case $q=2$, and in the case $k=2$. They conjectured that the bound is sharp for every prime power $q$ and every positive integer $k$. In this work quickly establish the truth of this conjecture. We provide two proofs, each employing different construction techniques. The first relies on the geometric view of linear codes as systems of projective points. The second approach is purely algebraic. We establish some lower bounds on the length of codes that satisfy the conjecture, and the length of the new codes constructed here are discussed.Comment: 19 page

n-Dimensional Optical Orthogonal Codes, Bounds and Optimal Constructions

We generalized to higher dimensions the notions of optical orthogonal codes. We establish uper bounds on the capacity of general $n$-dimensional OOCs, and on specific types of ideal codes (codes with zero off-peak autocorrelation). The bounds are based on the Johnson bound, and subsume many of the bounds that are typically applied to codes of dimension three or less. We also present two new constructions of ideal codes; one furnishes an infinite family of optimal codes for each dimension $n\ge 2$, and another which provides an asymptotically optimal family for each dimension $n\ge 2$. The constructions presented are based on certain point-sets in finite projective spaces of dimension $k$ over $GF(q)$ denoted $PG(k,q)$.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1702.0645

On the Weights of General MDS Codes

The weight spectra of MDS codes of length $n$ and dimension $k$ over the arbitrary alphabets are studied. For all $q$-ary MDS codes of dimension $k$ containing the zero codeword, it is shown that all $k$ weights from $n$ to $n-k+1$ are realized. The remaining case $n=q+k-1$ is also determined. Additionally, we prove that all binary MDS codes are equivalent to linear MDS codes. The proofs are combinatorial, and self contained

A note on full weight spectrum codes

A linear $[n,k]_q$ code $C$ is said to be a full weight spectrum (FWS) code if there exist codewords of each nonzero weight less than or equal to $n$. In this brief communication we determine necessary and sufficient conditions for the existence of linear $[n,k]_q$ full weight spectrum (FWS) codes. Central to our approach is the geometric view of linear codes, whereby columns of a generator matrix correspond to points in $PG(k-1,q)$

MWS and FWS Codes for Coordinate-Wise Weight Functions

A combinatorial problem concerning the maximum size of the (hamming) weight set of an $[n,k]_q$ linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those $[n,k]_q$ codes with the same weight set as $\mathbb{F}_q^n$ are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily ``short", whereas MWS codes are necessarily ``long". For fixed $k,q$ the values of $n$ for which an $[n,k]_q$-FWS code exists are completely determined, but the determination of the minimum length $M(H,k,q)$ of an $[n,k]_q$-MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on $n$ for which an FWS code exists, and bounds on $n$ for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on $M(\mathcal{L},k,q)$ (the minimum length of Lee MWS codes), and pose the determination of $M(\mathcal{L},k,q)$ as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes.Comment: 17 page

Age-related gait standards for healthy children and young people: the GOS-ICH paediatric gait centiles

Objective To develop paediatric gait standards in healthy children and young people. Methods This observational study builds on earlier work to address the lack of population standards for gait measurements in children. Analysing gait in children affected by neurological or musculoskeletal conditions is an important component of paediatric assessment but is often confounded by developmental changes. The standards presented here do not require clinician expertise to interpret and offer an alternative to developmental tables of normalised gait data. Healthy children aged 1-19 years were recruited from community settings in London and Hertfordshire, U.K. The GAITRite ® walkway was used to record measurements for each child for velocity, cadence, step length, base of support, and stance, single and double support (as percentage of gait cycle). We fitted generalized linear additive models for location, scale and shape (gamlss). Results We constructed percentile charts for seven gait variables measured on 624 (321 males) contemporary healthy children using gamlss package in R. A clinical application of gait standards was explored. Conclusion Age-related, gender-specific standards for seven gait variables were developed and are presented here. They have a familiar format and can be used clinically to aid diagnoses, and to monitor change over time for both medical therapy and natural history of the condition. The clinical example demonstrates the potential of the GOS-ICH Paediatric Gait Centiles (GOS-ICH PGC) to enable meaningful interpretation of change in an individual’s performance, and describes characteristic features of gait from a specific population throughout childhood.Peer reviewedFinal Accepted Versio

THE SYMMETRY ANGLE IDENTIFIES LESS CLINICALLY RELEVANT INTER-LIMB ASYMMETRIES THAN THE SYMMETRY INDEX IN HEALTHY ADULTS

There are several methods for calculating inter-limb symmetry, an inter-limb difference ≥15% has been suggested as an indicator of sporting injury risk. The purpose of this study was to compare three common methods for determining symmetry: the Symmetry Index (percentage difference; SI) when referenced to the left limb (SILeft) or the average of both limbs (SIAverage), and the Symmetry Angle (vector difference; SA). 15 recreationally active participants completed a sprint protocol on a non-motorised treadmill. Accelerometers were positioned on both tibias to measure peak resultant acceleration (PRA). The SA identified less clinically relevant PRA inter-limb asymmetries than the SI in healthy adults. Once an appropriate level of asymmetry as measured by the SA is determined, this may help to more correctly identify asymmetry in athletes and patients than the SI

Measurement of lower-limb asymmetry in professional rugby league: a technical note describing the use of inertial measurement units

Background. Quantifying lower-limb load and asymmetry during team sport match play may be important for injury prevention and understanding performance. However, current analysis methods of lower-limb symmetry during match-play employ wearable microtechnology that may not be best suited to the task. A popular microtechnology is global positioning systems (GPS), which are torso worn. The torso location, and the summary workload measures calculated by GPS are not suited to the calculation of lower-limb load. Instead, research grade accelerometers placed directly on the lower-limb may provide better load information than GPS. This study proposes a new technique to quantify external mechanical load, and lower-limb asymmetry during on-field team sport play using inertial measurement units. Methods. Four professional rugby league players (Age: 23.4 ± 3.1 years; Height: 1.89 ± 0.05 m; Mass: 107.0 ± 12.9 kg) wore two accelerometers, one attached to each foot by the boot laces, during match simulations. Custom Matlab (R2017b, The Mathworks Inc, Natick, MA) code was used to calculate total time, area under the curve (AUC), and percentage of time (%Time) spent in seven acceleration categories (negative to very high, <0 g to >16 g), as well as minimum and maximum acceleration during match simulations. Lower-limb AUC and %Time asymmetry was calculated using the Symmetry Angle Equation, which does not require normalization to a reference leg. Results. The range of accelerations experienced across all participants on the left and right sides were 15.68–17.53 g, and 16.18–17.69 g, respectively. Clinically significant asymmetry in AUC and %Time was observed for all but one participant, and only in negative (<0 g) and very high accelerations (>16 g). Clinically significant AUC differences in very high accelerations ranged from 19.10%–26.71%. Clinically significant %Time differences in negative accelerations ranged from 12.65%–25.14%, and in very high accelerations from 18.59%–25.30%. All participants experienced the most AUC at very low accelerations (2–4 g), and the least AUC at very high accelerations (165.00–194.00 AU vs. 0.32–3.59 AU). The %Time results indicated that all participants spent the majority of match-play (73.82–92.06%) in extremely low (0–2 g) to low (4–6 g) acceleration intensities, and the least %Time in very high accelerations (0.01%–0.05%). Discussion. A wearable located on the footwear to measure lower-limb load and asymmetry is feasible to use during rugby league match-play. The location of the sensor on the boot is suited to minimize injury risk occurring from impact to the sensor. This technique is able to quantify external mechanical load and detect inter limb asymmetries during match-play at the source of impact and loading, and is therefore likely to be better than current torso based methods. The results of this study may assist in preparing athletes for match-play, and in preventing injury

The Demands of Professional Rugby League Match-Play: a Meta-analysis

Background Rugby league is a collision sport, where players are expected to be physically competent in a range of areas, including aerobic fitness, strength, speed and power. Several studies have attempted to characterise the physical demands of rugby league match-play, but these studies often have relatively small sample sizes based on one or two clubs, which makes generalisation of the findings difficult. Therefore, the aim of this review was to synthesise studies that investigated the physical demands of professional rugby league match-play. Methods SPORTDiscus, CINAHL, MEDLINE (EBSCO) and Embase (EBSCO) databases were systematically searched from inception until October 2018. Articles were included if they (1) recruited professional rugby league athletes aged ≥ 18 years and (2) provided at least one match-play relevant variable (including playing time, total and relative distance, repeat high-intensity efforts (RHIE), efforts per RHIE, accelerations and decelerations, total and relative collisions). Meta-analyses were used to provide pooled estimates ± 95% confidence intervals. Results A total of 30 studies were included. Pooled estimates indicated that, compared to adjustables and backs, forwards have less playing time (− 17.2 ± 5.6 and − 25.6 ± 5.8 min, respectively), cover less ‘slow-speed’ (− 2230 ± 735 and − 1348 ± 655 m, respectively) and ‘high-speed’ distance (− 139 ± 108 and − 229 ± 101 m, respectively), but complete more relative RHIEs (+ 0.05 ± 0.05 and + 0.08 ± 0.04 per minute, respectively), and total (+ 12.0 ± 8.1 and + 12.8 ± 7.2 collisions, respectively) and relative collisions (+ 0.32 ± 0.22 and + 0.41 ± 0.22 collisions per minute, respectively). Notably, when the distance was expressed relative to playing time, forwards were not different from adjustables and backs in slow-speed (P ≥ 0.295) and high-speed (P ≥ 0.889) relative distance. The adjustables and backs subgroups were similar in most variables, except playing time (shorter for adjustables, − 8.5 ± 6.2 min), slow-speed distance (greater for adjustables, + 882 ± 763 m) and total relative distance (greater for adjustables, + 11.3 ± 5.2 m·min−1). There were no significant differences between positional groups for efforts per RHIE, accelerations and decelerations (P ≥ 0.745). Conclusions These results indicate the unique physical demands of each playing position and should be considered by strength and conditioning and tactical coaches when planning for professional rugby league performance

Understanding and supporting block play: video observation research on preschoolers’ block play to identify features associated with the development of abstract thinking

This article reports on a study conducted to investigate the development of abstract thinking in preschool children (ages from 3 years to 4 years old) in a nursery school in England. Adopting a social influence approach, the researcher engaged in 'close listening' to document children's ideas expressed in various representations through video observation. The aim was to identify behaviours connected with features of the functional dependency relationship – a cognitive function that connects symbolic representations with abstract thinking. The article presents three episodes to demonstrate three dominating features, which are i) child/child sharing of thinking and adult and child sharing of thinking; ii) pause for reflection; and iii) satisfaction as a result of self-directed play. These features were identified as signs of learning, and were highlighted as phenomena that can help practitioners to understand the value of quality play and so provide adequate time and space for young children and plan for a meaningful learning environment. The study has also revealed the importance of block play in promoting abstract thinking. Keywords: abstract thinking; functional dependency relationship; social influence approach; block play; preschool; video observation; qualitative researc