On 12/15/2013 06:46 AM, Peter Mueller wrote:
The function delsarte_bound_hamming_space(n, d, q, isinteger=False, return_data=False, solver='PPL') offers the option isinteger=True. As the doc doesn't tell it, I got a little curious what is assumed to be integral. Looking at the implementation it turns out that the distance distribution is assumed to consist of integers. However, for non-linear codes these numbers rarely are integral!
I think isinteger probably makes sense for the delsarte_bound_additive_hamming_space() function.
Checking the bounds obtained by this didn't produce anything which contradicts known lower bounds, but it improves quite a few known upper bounds in Agrell's and Brouwer's tables (modulo the fact that the MIP solvers are based on floating point LP solvers and thus don't give proven results.)
The LP solver used should give exact results and not have any floating point problems. See http://trac.sagemath.org/ticket/12533
So I seriously doubt that the isinteger=True is based on a valid mathematical theorem, or is there some extension of Delsarte's Theorem which allows to assume that the distance distribution in an optimal code consists of integers? -- Peter Mueller
I am not sure why there would be improvements to the tables. Someone would have noticed such improvements earlier, right? The result obtaind from allowing isinteger=True should be an upper bound to the actual maximization problem. This is because putting isinteger=True imposes more constraints on the variables (the distance distribution) and so the result obtained from setting this to true should be at least as large as the result obtained from setting this to false (since the constraint space is larger in the latter case).
Can you perhaps give an example where setting isinteger=True gives a bigger number than when setting isinteger=False?
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