A New Bound for the Minimum Distance of a Cyclic Code From Its Defining Set

A New Bound for Cyclic Codes Beating the Roos Bound

“…However, from the computed codes it appears that bound C is tighter than the Roos bound overall. Although the BS bound sometimes beats the Roos bound, in the majority of computed cases the Roos bound is better, as reported in [BS06] and checked by us. Bound C is the first polynomial-time bound outperforming the Roos bound on a significant sample of codes.…”

“…This part presents our notation and some preliminary results following mainly [MS77], [BS07] and [BS06].…”

“…To check if a set of vectors in U n is linearly independent, we use the socalled "singleton procedure" (see [BS06], [BS07], [PS03]). Any ordered multiset of t rows of length n with t ≤ n forms a matrix M t ∈ U t×n .…”

“…, m, that is, we take the row with the primary pivot in the first position and its shifts up to the (m − 1)−th shift included. We collect these rows in submatrix T 1 and we note that they are clearly linearly independent, applying the singleton procedure (see [BS06], Lemma 3.2). Note that T 1 and T 2 have no common rows.…”

“…There are many lower bounds for the distance of cyclic codes that use some particular patterns in the set of zeros of the generator polynomial, as for example the Bose-Chaudhuri-Hockenheim (BCH) bound [BRC60b], the Hartmann-Tzeng (HT) bound [HT72], the Betti-Sala (BS) bound [BS06] and the Roos bound [Roo83]. We focus on this kind of bounds.…”