Evaluation codes from order domain theory

Abstract

The celebrated Feng-Rao bound estimates the minimum distance of codes defined by means of their parity check matrices. From the Feng-Rao bound it is clear how to improve a large family of codes by leaving out certain rows in their parity check matrices. In this paper we derive a simple lower bound on the minimum distance of codes defined by means of their generator matrices. From our bound it is clear how to improve a large family of codes by adding certain rows to their generator matrices. The new bound is very much related to the Feng-Rao bound as well as to Shibuya and Sakaniwa's bound in [T. Shibuya, K. Sakaniwa, A dual of well-behaving type designed minimum distance, IEICE Trans. Fund. E84-A (2001) 647-652]. Our bound is easily extended to deal with any generalized Hamming weights. We interpret our methods into the setting of order domain theory. In this way we fill in an obvious gap in the theory of order domains.

On the evaluation codes given by simple $$\delta $$ δ -sequences

“…The following result summarizes the known bounds for the minimum distances, d, of the primary and dual codes defined by w and φ (see [12,13,31,22,3]). …”

“…The above mentioned primary family of codes has been studied in a recent paper [3]. There were also introduced the improved primary codes and an order-type bound for primary codes.…”

“…Γ generates an additive well-ordered semigroup (with respect to the lexicographic order < in Z 2 with (0, 1) < (1, 0)), which will be denoted as S Γ . As an example, we can say that Γ = { (18,9), (6,3), (4,2), (1, 1)} is a δ-sequence determined by Γ * = {45, 15, 10, 3}. Indeed, the pair (m 2 , e 2 ) for Γ * is (3 · 10 − 3, 5) = (27,5) and ⟨a 1 ; a 2 , a 3 ⟩ = ⟨5; 2, 2⟩, so t = 3, y −1 = (0, 1), y 0 = (1, 0) = y t−3 = (A ′ , B ′ ) and (A, B) = 2(1, 0) + (0, 1) = (2, 1).…”

“…For instance, in the above example, gcd ((18, 9), (6, 3)) = (6, 3) because by the extended Euclidean algorithm (18, 9) = 3(6, 3) + (0, 0), gcd ((18, 9), (6,3), (4, 2)) = (2, 1) and 3 = (6, 3)/(2, 1). The construction of Γ proves that these values n i coincide with those n * i for Γ * since each…”

“…However, under some restrictions and when A is an affine algebra, in [3] it is described a way to do it. We will use this way to give a bound on the minimum distance of the primary codes given by simple δ-sequences.…”