AI 中文总结
本文研究二元近完备覆盖码，修正原Van Wee界以纳入最小距离，确定R=2、3的此类码等价于构造的码，且R≥3时此类码数量有限。
AI 中文摘要
我们研究（二元）近完备覆盖码，这类码能等号达到Van Wee界，是近完备纠错码的覆盖对应物，后者能等号达到Johnson界。覆盖半径R=1的这类码已被完全分类。我们证明R≥2的码无法等号达到原Van Wee界，因为该界未考虑码的最小距离；我们修正该界以纳入最小距离，并构造了一些近完备覆盖码。通过证明这类码的若干结构性质，我们证明所有R=2、3的近完备覆盖码必等价于所构造的码；还证明对任意R≥3，近完备覆盖码至多有有限个。
英文摘要
We study (binary) nearly-perfect covering codes, which are codes that attain the Van Wee bound with equality. They act as the covering counterparts to nearly-perfect error-correcting codes, which attain the Johnson bound with equality. These codes have been completely classified for covering radius $R=1$. We prove that no code with $R\geq 2$ can attain the original Van Wee bound with equality, since it omits the dependence on the minimum distance of the code. We refine the bound to account for the minimum distance and show some nearly-perfect covering codes. By proving some structural properties of such codes, we prove all nearly-perfect covering codes with $R=2,3$ must be equivalent to the codes we showed. We also prove that for any $R\geq 3$, there are at most a finite number of nearly-perfect covering codes.
Comments26 pages
