A new class of optimal linear codes with flexible parameters

MacDonald codes over the ring $${\mathbb {F}}_{p}+v{\mathbb {F}}_{p}+v^2{\mathbb {F}}_{p}$$

“…In Luo et al (2018), the authors obtained a class of linear codes with two weights over F q . They also employed these linear codes to construct association schemes.…”

“…Definition 3 (Luo et al 2018) Let X be a finite set with the cardinality of X greater than 2, L 0 = {(a, a)| a ∈ X }, L i = {(a, b)| a, b ∈ X }, i = 1, 2, . .…”

“…In Luo et al (2018), the authors give a necessary and sufficient condition for constructing a 2-class association scheme from a linear code with parameters [n, 2].…”

“…Lemma 6 (Luo et al 2018) Assume that C is an [n, 2] linear code over F q with nonzero weights w 1 and w 2 . Let c 1 , c 2 be two codewords of C such that c 1 , c 2 are linear independent and w(c 1 ) = w(c 2 ) = w 1 .…”

“…Especially, they provide a framework to study codes and designs. Luo et al (2018) constructed a class of linear codes with two weights over F q by linear codes over the finite chain ring F q + uF q with u 2 = 0. They also employed these linear codes to construct association schemes.…”