On Bounded Weight Codes

Christine Bachoc (1) , Venkat Chandar , Gerard Cohen (2) , Patrick Sole (3) , Aslan Tchamkerten

1 IMB - Institut de Mathématiques de Bordeaux
2 Ministère de la Culture et de la communication, service du Livre et de la lecture, bureau du Patrimoine
3 Télécom ParisTech

The maximum size of a binary code is studied as a function of its length N, minimum distance D, and minimum codeword weight W. This function B(N,D,W) is first characterized in terms of its exponential growth rate in the limit as N tends to infinity for fixed d=D/N and w=W/N. The exponential growth rate of B(N,D,W) is shown to be equal to the exponential growth rate of A(N,D) for w <= 1/2, and equal to the exponential growth rate of A(N,D,W) for 1/2< w <= 1. Second, analytic and numerical upper bounds on B(N,D,W) are derived using the semidefinite programming (SDP) method. These bounds yield a non-asymptotic improvement of the second Johnson bound and are tight for certain values of the parameters.

Domaines: Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]

Christine Bachoc, Venkat Chandar, Gerard Cohen, Patrick Sole, Aslan Tchamkerten. On Bounded Weight Codes. IEEE Transactions on Information Theory, 2011, 57 (10), pp.6780-6787. ⟨hal-00911674⟩