Characterization of the sets L(d,r,i)

Determine the form of the sets L(d,r,i) arising in the generalized Sudan list-decoding algorithm for hyperbolic codes, in order to compute their sizes and thereby assess the algorithm’s decoding radius.

Background

The paper defines L(d,r,i) as the set of exponent vectors a for which a+iH is contained in H_r, where H defines the hyperbolic code Hyp_q(d,m) and H_r defines Hyp_q(r+1,m). These sets determine the number of unknown coefficients available when constructing the interpolation relation used by the Geil–Matsumoto generalized Sudan list-decoding algorithm.

The authors explain that computing the sizes of the sets L(d,r,i) is necessary for determining the number of errors that the algorithm can correct. They subsequently derive bounds and, in the two-variable case, exact descriptions for these sets, while leaving their general form unresolved.