Poset code[1] 

Poset codes are quantified with respect to the poset metric [2]. This metric is based on a partial ordering \(\leq\) on subsets of \([n]=\{1,2,\cdots,n\}\) and the notion of ideals generated by the support of an element of a \(q\)-ary string. An ideal \(\langle \text{supp}(x) \rangle\) generated by \(x\in GF(q)^n\) contains all subsets of \([n]\) that are less than or equal to the subset in the support of \(x\) in the partial ordering. The poset metric between two strings \(x,y\) is then the cardinality of the ideal generated by the difference between the supports of \(x\) and \(y\), \(d_P(x,y) = |\langle \text{supp}(x-y) \rangle|\).

Generalizations of various bounds for ordinary \(q\)-ary codes have been developed for poset codes, including generalizations of MacWilliams identities [3]; see [2].