偏序集上具有扩张性质的线性码
Linear codes on posets with extension property
摘要：
我们研究偏序类Hamming空间中满足扩展性质的线性和加法码，这意味着理想的自同构扩展到偏序集的自同构。代码自然地描述的翻译协会计划，起源于组的线性等距的空间。我们解决问题的对偶和不变量的代码，建立一个连接的对偶协会计划和计划定义的对偶偏序集（他们是同构的，当且仅当偏序集是自对偶）。我们进一步讨论了在偏序集的情况下，扮演码的重量枚举数的角色的不变量。在正规根树的情况下，这种不变量与树同构的经典问题有关。我们还研究了这些不变量是否在偏序集的标准运算下保持不变的问题，如序数和等。
英文摘要：
We investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of translation association schemes that originate from the groups of linear isometries of the space. We address questions of duality and invariants of codes, establishing a connection between the dual association scheme and the scheme defined on the dual poset (they are isomorphic if and only if the poset is self-dual). We further discuss invariants that play the role of weight enumerators of codes in the poset case. In the case of regular rooted trees such invariants are linked to the classical problem of tree isomorphism. We also study the question of whether these invariants are preserved under standard operations on posets such as the ordinal sum and the like.
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