Golay and other box codes

The (24,12;8) extended Golay Code can be generated as a 6 x 4 binary matrix from the (15,11;3) BCH-Hamming Code, represented as a 5 x 3 matrix, by adding a row and a column, both of odd or even parity. The odd-parity case provides the additional 12th dimension. Furthermore, any three columns and five rows of the 6 x 4 Golay form a BCH-Hamming (15,11;3) Code. Similarly a (80,58;8) code can be generated as a 10 x 8 binary matrix from the (63,57;3) BCH-Hamming Code represented as a 9 x 7 matrix by adding a row and a column both of odd and even parity. Furthermore, any seven columns along with the top nine rows is a BCH-Hamming (53,57;3) Code. A (80,40;16) 10 x 8 matrix binary code with weight structure identical to the extended (80,40;16) Quadratic Residue Code is generated from a (63,39;7) binary cyclic code represented as a 9 x 7 matrix, by adding a row and a column, both of odd or even parity

Improved Correlation between the Variability and Peak Luminosity of Gamma-Ray Bursts

A new procedure for smoothing a gamma-ray burst (GRB) lightcurve and calculating its variability is presented. Applying the procedure to a sample of 25 long GRBs, we have obtained a very tight correlation between the variability and the peak luminosity. The only significant outlier in the sample is GRB 030329. With this outlier excluded, the data scatter is reduced by a factor of \sim 3 compared to that of Guidorzi et al. (2005), measured by the deviation of fit. Possible causes for the outlier are discussed.Comment: 9 pages, including 7 figure. Expanded as requested by referee. An error in Fig. 1 was corrected (the two 1-sigma lines in the old version were wrong). Accepted for publication by MNRA

The Mathieu group M-12 and its pseudogroup extension M-13

We study a construction of the Mathieu group M-12 using a game reminiscent of Loyd's "15-puzzle." The elements of M-12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There is a natural extension to a "pseudogroup" M-13 acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, akin to that induced by a Cayley graph, on both M-12 and M-13. We develop these results, and extend them to the double covers and automorphism groups of M-12 and M-13, using the ternary Golay code and 12 x 12 Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.Mathematic

Early Science with the Large Millimeter Telescope: COOL BUDHIES I - a pilot study of molecular and atomic gas at z~0.2

An understanding of the mass build-up in galaxies over time necessitates tracing the evolution of cold gas (molecular and atomic) in galaxies. To that end, we have conducted a pilot study called CO Observations with the LMT of the Blind Ultra-Deep H I Environment Survey (COOL BUDHIES). We have observed 23 galaxies in and around the two clusters Abell 2192 (z = 0.188) and Abell 963 (z = 0.206), where 12 are cluster members and 11 are slightly in the foreground or background, using about 28 total hours on the Redshift Search Receiver (RSR) on the Large Millimeter Telescope (LMT) to measure the $^{12}$CO J = 1 --> 0 emission line and obtain molecular gas masses. These new observations provide a unique opportunity to probe both the molecular and atomic components of galaxies as a function of environment beyond the local Universe. For our sample of 23 galaxies, nine have reliable detections (S/N$\geq$3.6) of the $^{12}$CO line, and another six have marginal detections (2.0 < S/N < 3.6). For the remaining eight targets we can place upper limits on molecular gas masses roughly between $10^9$ and $10^{10} M_\odot$. Comparing our results to other studies of molecular gas, we find that our sample is significantly more abundant in molecular gas overall, when compared to the stellar and the atomic gas component, and our median molecular gas fraction lies about $1\sigma$ above the upper limits of proposed redshift evolution in earlier studies. We discuss possible reasons for this discrepancy, with the most likely conclusion being target selection and Eddington bias.Comment: MNRAS, submitte

The Mathieu group M12 and the M13 game

This is the author's accepted manuscript

Golay encoding/decoding via BCH-Hamming

AbstractThe (24, 12;8) extended Golay Code can be generated as a 6 × 4 binary matrix from the (15, 11;3) BCH-Hamming Code, represented as a 5 × 3 matrix, by adding a row and column, both of odd or even parity. The odd parity case provides the additional 12th dimension. Furthermore, any three columns and five rows of the 6 × 4 Golay form a permuted BCH- Hamming (15, 11;3) Code. Using the single-error correction of the BCH-Hamming Code, it is possible to decode in a primitive combinatoric fashion all three error patterns. Single- and double-error correction is almost immediate

The mediating effect of task presentation on collaboration and children's acquisition of scientific reasoning

There has been considerable research concerning peer interaction and the acquisition of children's scientific reasoning. This study investigated differences in collaborative activity between pairs of children working around a computer with pairs of children working with physical apparatus and related any differences to the development of children's scientific reasoning. Children aged between 9 and 10 years old (48 boys and 48 girls) were placed into either same ability or mixed ability pairs according to their individual, pre-test performance on a scientific reasoning task. These pairs then worked on either a computer version or a physical version of Inhelder and Piaget's (1958) chemical combination task. Type of presentation was found to mediate the nature and type of collaborative activity. The mixed-ability pairs working around the computer talked proportionally more about the task and management of the task; had proportionally more transactive discussions and used the record more productively than children working with the physical apparatus. Type of presentation was also found to mediated children's learning. Children in same ability pairs who worked with the physical apparatus improved significantly more than same ability pairs who worked around the computer. These findings were partially predicted from a socio-cultural theory and show the importance of tools for mediating collaborative activity and collaborative learning