On combinatorial searches for designs and codes

• Main Author: Houghten, Sheridan
• Format: Others
• Published: 1999
• Online Access: http://spectrum.library.concordia.ca/794/1/NQ43587.pdf; Houghten, Sheridan <http://spectrum.library.concordia.ca/view/creators/Houghten=3ASheridan=3A=3A.html> (1999) On combinatorial searches for designs and codes. PhD thesis, Concordia University.

Searches for designs and codes may be divided into two broad categories: those which attempt to prove the existence of a design or code with a given set of parameters, and those which attempt to enumerate designs or codes with a given set of parameters. In this thesis we consider computer searches for several designs and one code. The also consider computational techniques which one may generally employ in such searches. A 2-(28, 12, 11) design is both quasi-derived and quasi-symmetric, with intersection numbers a = 4 and b = 6. We consider the problem of embedding such designs with an automorphism of order 7 without fixed points or blocks, as derived designs in a symmetric 2-(64, 28, 12) design. We find that four 2-(28, 12, 11) designs could not be embedded, thus providing the first examples of quasi-derived quasi-symmetric designs which are not derived designs. The smallest case with the parameters k = 6 and n = 1 for which it is not known if a ( v , 6, 1) design exists is the case v = 46. One may partition the search for a (46, 6, 1) design into two cases. A search assuming the first of these cases revealed no designs. We examine the remaining case, presenting detailed information on how one may structure a search, and present estimates of the time required to complete the search. The largest minimum weight of a self-dual doubly-even binary ( n, k, d ) code is d = 4 [ n /24] + 4. Of such codes with length divisible by 24, the Golay Code is the only (24, 12, 8) code, the Extended Quadratic Residue Code is the only known (48, 24, 12) code, and there is no known (72, 36, 16) code. For any non-zero weight w of such a code, the codewords of weight w form a 5-design. One may partition the search for a (48, 24, 12) self-dual doubly-even code into 3 cases. A search assuming one of the cases found only the Extended Quadratic Residue Code itself. We consider the remaining 2 cases, giving information on how a search may be structured for each of these cases, and present estimates of the search size and time. Estimation is of great importance in these types of searches, as it not only indicates if a search is feasible with the given resources, but also may indicate necessary areas of improvement in our algorithms. We examine methods of obtaining estimates of the size of a search and of the time required to complete a search, including a demonstration of a conversion of the former type of estimate to the latter.