Calculating the minimum distance of linear block codes via Integer Programming

Minimum-Variance Importance-Sampling Bernoulli Estimator for Fast Simulation of Linear Block Codes over Binary Symmetric Channels

“…The result of the Theorem 5 suggests a more computationally efficient IS-MC simulation algorithm that improves the basic algorithm derived in the previous sub-section. In fact, under the assumptions of the Theorem 5, theq given by (23) does not depend on the current specific cross-over probability p of the channel being simulated. Then, the same set of generated samples withq can be used to calculate the estimate of the WER at different SNRs.…”

“…Proof: The proof is given in Appendix D. A notable consequence of Theorem 5 is the independence ofq from the cross-over probability p (which in turn depends on the SNR), therefore leading to an SNR-invariant IS-MC simulation. In this case estimation of WERs for a whole range of SNRs can be obtained by running one IS-MC simulation with a BSC with parameterq given by (23), in the place of one simulation for each SNR. Thus the whole performance curve WER versus SNR can be obtained with a dramatic reduction of the number of samples to be generated.…”

“…Obviously, ifq 0 is close to the optimal solution then a small number of iterations is required. If the number of correctable errors t is known then a possible choice could be theq given by (23), even though for np ≪ 1 a more computationally efficient simulation algorithm is possible, as it will be shown in the next section. An alternative choice can be made by observing that in a typical scenario the algorithm is run to draw a performance curve as function of the SNRs or the cross-over probability p. One can use theq estimated with the simulation at the previous SNR as starting probability for the current SNR, i.e.q 0 (SNR i + ∆SNR) =q l (SNR i ), since for relatively small ∆SNR the new optimalq is expected to be in the neighborhood of the previousq.…”

“…Note also that for short codes the assumption np ≪ 1 holds for a large range of SNRs and then (23) is valid for values of p of interest, while for long codes the same assumption holds only for high SNRs and (23) may not be useful in practice.…”

“…For this class of codes the method has been improved in [21] and [22]. More recently, integer programming methods have been used to calculate either the true minimum distance or an upper bound [23]. Alternatively, a branch-and-cut algorithm for finding the minimum distance of linear block codes has been proposed in [24].…”