Linear Programming Bounds on the Kissing Number of q-ary Codes

Abstract

We use linear programming (LP) to derive upper and lower bounds on the “kissing number” $A_{d}$ of any q-ary linear code C with distance distribution frequencies $A_{i}$, in terms of the given parameters $[n,\ k,\ d]$. In particular, a polynomial method gives explicit analytic bounds in a certain range of parameters, which are sharp for some low-rate codes like the first-order Reed-Muller codes. The general LP bounds are more suited to numerical estimates. Besides the classical estimation of the probability of decoding error and of undetected error, we outline recent applications in hardware protection against side-channel attacks using code-based masking countermeasures, where the protection is all the more efficient a s the kissing number is low.