Abstract Let p be a prime number, in this paper, we investigate the structures of all constacyclic codes of length p s over the ring R u 2 , v 2 , p m = F p m [ u , v ] ∕ 〈 u 2 , v 2 , u v − v u 〉 . The units of the ring R u 2 , v 2 , p m can be divided into following five forms: α , λ 1 = α + δ 1 u v , λ 2 = α + γ v + δ u v , λ 3 = α + β u + δ u v , λ 4 = α + β u + γ v + δ u v , where α , β , γ , δ 1 ∈ F p m ∗ and δ ∈ F p m . We obtain the algebraic structures of all constacyclic codes of length p s over R u 2 , v 2 , p m , except ( α + δ 1 u v ) -constacyclic codes, in terms of their polynomial generators and also find the number of codewords in each of these constacyclic codes. The number of constacyclic codes and duals of constacyclic codes corresponding to the units λ 2 , λ 3 and λ 4 are determined. We also provide examples to illustrate our results, which include several optimal codes.

中文翻译：

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在 Fpm[u,v]∕〈u2,v2,uv−vu> 上长度为 ps 的恒循环码

摘要 设 p 为素数，本文研究了环 R u 2 , v 2 , pm = F pm [ u , v ] ∕ < u 2 , v 2 上所有长度为 ps 的恒循环码的结构， uv - vu 〉 。环的单位 R u 2 , v 2 , pm 可分为以下五种形式： α , λ 1 = α + δ 1 uv , λ 2 = α + γ v + δ uv , λ 3 = α + β u + δ uv , λ 4 = α + β u + γ v + δ uv ，其中 α , β , γ , δ 1 ∈ F pm ∗ 和 δ ∈ F pm 。我们获得了 R u 2 , v 2 , pm 上所有长度为 ps 的恒环码的代数结构，除了 (α + δ 1 uv ) - 恒环码，根据它们的多项式生成器，并找到每个中的码字数这些恒循环码。确定与单位λ 2 、λ 3 和λ 4 对应的恒环码的数量和恒环码的对偶数。