Disproof of the Shamai-Laroia conjecture

We prove that the conjectured Shamai-Laroia lower bound for the maximum achievable rate in the intersymbol interference channel does not hold. It is shown that in the low SNR regime, the opposite to the conjecture is true, for all ISI channels and essentially all input distributions. Using insights obtained from this proof, we construct a clear counterexample to the conjecture.