基于小域上负循环码与重根循环码的量子局部可修复码
Quantum Locally Repairable Codes from Negacyclic and Repeated-Root Cyclic Codes over Small Fields
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中文总结 AI 辅助
本文研究小域上经典常循环码经CSS构造的量子局部可修复码，证明仅负循环和重根循环码可用于该构造，给出局部性与纯度判据，得到无限纯qLRCs族及首批二元重根qLRCs等新参数。
中文摘要 AI 辅助
量子局部可修复码（qLRCs）由Golowich和Guruswami近期提出，可从少量其他量子位（qudit）恢复任意单个量子位擦除错误，目前已知的多数构造都需要大字母表。本文系统研究了通过CSS构造，从q∈{2,3,4,5,7}的小域F_q上的经典常循环码得到的qLRCs。首先，我们证明仅当λ²=1时才存在非零的含对偶λ-常循环码，因此负循环码和（重根）循环码涵盖了通过常循环途径构造qLRCs的所有可能。其次，我们证明常循环码的局部性等于其对偶码的最小距离减一，并给出了所得量子码的简单纯度判据。第三，我们证明分裂由μ_{-1}给出的奇型Duadic码可生成纯qLRCs；将其特例化为素长度p≡3 mod 4的q元二次剩余码，可得到一类具有无界最小距离和可认证局部性的无限纯qLRCs族。最后，通过具体计算，我们对中等长度的循环码、负循环码和重根循环码构造的qLRCs进行了分类，其中包含了首批由重根循环码构造的二元qLRCs，以及许多循环码无法达到的参数组。
英文摘要
Quantum locally recoverable codes (qLRCs), introduced recently by Golowich and Guruswami, allow any single-qudit erasure to be recovered from a small set of other qudits. Most known constructions require a large alphabet. We systematically investigate qLRCs obtained, via the CSS construction, from classical constacyclic codes over small fields $\Ff_q$ with $q\in\{2,3,4,5,7\}$. First, we prove that a nonzero dual-containing $λ$-constacyclic code exists only when $λ^2=1$, so that negacyclic and (repeated-root) cyclic codes exhaust the constacyclic route to qLRCs. Second, we show that the locality of a constacyclic code equals the minimum distance of its dual minus one, and we give a simple purity criterion for the resulting quantum codes. Third, we show that odd-like duadic codes whose splitting is given by $μ_{-1}$ yield pure qLRCs; specializing to $q$-ary quadratic residue codes of prime length $p\equiv 3 \pmod 4$ gives an infinite family of pure qLRCs with unbounded minimum distance and certified locality. Finally, by means of concrete computations, we obtain a classification of qLRCs from cyclic, negacyclic, and repeated-root cyclic codes of moderate lengths, which contains the first binary qLRCs from repeated-root cyclic codes and many parameter sets that cyclic codes cannot attain.
