The binary $k$-dimensional simplex code is known to be a $2^{k-1}$-batch code and is conjectured to be a $2^{k-1}$-functional batch code. Here, we offer a simple, constructive proof of a result that is "in between" these two properties. Our approach is to relate these properties to certain (old and new) additive problems in finite abelian groups. We also formulate a conjecture for finite abelian groups that generalizes the above-mentioned conjecture.

翻译：二进制 $k$- 维维度简单x 代码已知为 2 ⁇ k- 1} 美元批量代码, 并被推断为 2 ⁇ k- 1} 美元功能批量代码 。 在这里, 我们提供一个简单、 建设性的证据, 证明结果是“ 介于” 这两个属性之间 。 我们的方法是将这些属性与某些( 旧的和新的) 限制的亚伯尔语组的添加问题联系起来 。 我们还为上述假设的有限频谱组拟定一个猜想 。