Perfect Codes on Some Ordered Sets

Using the concept of codes on ordered sets introduced by Brualdi, Graves and Lawrence, we consider perfect codes on the ordinal sum of two ordered sets, the standard ordered sets and the disjoint sum of two chains. In the classical coding theory all perfect codes are completely described in terms of parameters (cf. [2, 3]). They are mathematically interesting structures. Brualdi et al. [1] recently introduced the notion of a code on an ordered set. In this paper we consider perfect codes on the ordinal sum of two ordered sets, the standard ordered sets and the disjoint sum of two chains. A chain is an ordered set in which every two elements are comparable and an antichain in which no two elements are comparable. Then we denote by n and n denote the chain and the antichain, respectively, on the set {1, 2, . . . , n}. Let P and Q be two disjoint ordered sets. The disjoint sum P + Q of P and Q is the ordered set on P ∪ Q such that x < y if and only if x, y ∈ P and x < y in P or x, y ∈ Q and x < y in Q, and the ordinal sum P ⊕ Q of P and Q is obtained from P + Q by adding the new relations x < y for all x ∈ P and y ∈ Q. Let Fq be a finite field with q = p (p a prime, d a positive integer). For u = (u1, . . . , un) ∈ Fq , the support of u and the Hamming weight of u are respectively given by Supp(u) = {i : 1 ≤ i ≤ n, ui 6= 0},