We obtain new linear programming (LP) and constructive bounds for the covering radius of binary orthogonal arrays of strength $2k$. Our LP bounds develop in two alternative scenarios. First, if a point $y \in F_2^n$, where the covering radius of some orthogonal array $C \subset F_2^n$ of strength $2k$ is realized, is such that the farthest point of $C$ to $y$ is not antipodal to $y$ we obtain a bound which is better than the Tiet{ä}v{ä}inen (or Fazekas-Levenshtein) bound for non-tight arrays (i.e., the cardinality strictly exceeds the Rao lower bound). Second, if all points where the covering radius is realized are such that their antipodes are in $C$, we obtain a bound which depends on the cardinality of $C$ and is again better whenever the orthogonal array is not tight. We further describe three infinite families of binary orthogonal arrays related to the duals of BCH, Melas, and Zetterberg codes. For these families, we derive lower bounds on the covering radius by applying techniques from algebraic curves over finite fields, while the improved linear programming methods developed in this paper provide upper bounds, leading in some cases to fairly close estimates.

翻译：我们获得了二元$2k$强度正交阵列覆盖半径的新的线性规划（LP）界及构造性界。我们的LP界在两种替代场景下得以发展。首先，若一点$y \in F_2^n$（其中强度为$2k$的正交阵列$C \subset F_2^n$的覆盖半径在此点实现）满足$C$中距$y$最远的点不是$y$的对极点，则对于非紧致阵列（即基数严格超过Rao下界的情况），我们得到的界优于Tiet{ä}v{ä}inen（或Fazekas-Levenshtein）界。其次，若所有实现覆盖半径的点均满足其对极点位于$C$中，则我们得到的界依赖于$C$的基数，且同样在正交阵列非紧致时更加优越。我们进一步描述了与BCH码、Melas码和Zetterberg码的对偶码相关的三个无穷二元正交阵列族。对于这些族，我们应用有限域上代数曲线技术推导了覆盖半径的下界，而本文发展的改进线性规划方法则提供了上界，在某些情形下得到了相当接近的估计值。