Upper Bound on List-Decoding Radius of Binary Codes

Consider the problem of packing Hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient Hamming space with multiplicity at most
$L$
. For odd
$L\ge 3$
, an asymptotic upper bound on the rate of any such packing is proved. The resulting bound improves the best known bound (due to Blinovsky’1986) for rates below a certain threshold. The method is a superposition of the linear-programming idea of Ashikhmin, Barg, and Litsyn (that was previously used to improve the estimates of Blinovsky for
$L=2$
) and a Ramsey-theoretic technique of Blinovsky. As an application, it is shown that for all odd
$L$
, the slope of the rate-radius tradeoff is zero at zero rate.