Conjectures and Known Properties of the Unknown Code
Does their exist a Type II [24k, 12k, 4k+4] code for some positive integer k >2? These are called extremal codes and are known to exist for k<3. Neal Sloane posed the question in 1973 for k≥3 which is still unsolved. This talk will focus on the first unknown case where k=3. Why is this code so elusive? Amazingly, there are a lot of known properties about this code despite the fact the code is not know to exist or not. Proving existence or nonexistence of this code would also answer a few other unknown mathematical problems, such as the existence of a 5-(72,16,78) t-design. The search itself has yielded empirical evidence that gives rise to many interesting conjectures.