In coding theory, codes are usually designed with a certain level of randomness to facilitate analysis and accommodate different channel conditions. However, the resulting random code constructed can be suboptimal in practical implementations. Represented by a bipartite graph, the Batched Sparse Code (BATS Code) is a randomly constructed erasure code that utilizes network coding to achieve near-optimal performance in wireless multi-hop networks. In the performance analysis in the previous research, it is implicitly assumed that the coded batches in the BATS code are independent. This assumption holds only asymptotically when the number of input symbols is infinite, but it does not generally hold in a practical setting where the number of input symbols is finite, especially when the code is constructed randomly. We show that dependence among the batches significantly degrades the code's performance. In order to control the batch dependence through graphical design, we propose constructing the BATS code in a structured manner. A hardware-friendly structured BATS code called the Cyclic-Shift BATS (CS-BATS) code is proposed, which constructs the code from a small base graph using light-weight cyclic-shift operations. We demonstrate that when the base graph is properly designed, a higher decoding rate and a smaller complexity can be achieved compared with the random BATS code.

翻译：在编码理论中，编码通常被设计为具有一定随机性，以便于分析并适应不同的信道条件。然而，由此构造的随机码在实际实现中可能并非最优。批量稀疏码（BATS码）是一种以二分图表示的随机构造的擦除码，它利用网络编码在无线多跳网络中实现接近最优的性能。在先前研究的性能分析中，隐含地假设了BATS码中的编码批次是相互独立的。这一假设仅在输入符号数量趋于无穷时渐近成立，但在输入符号数量有限的实际情况中通常并不成立，尤其是在编码为随机构造时。我们证明批次间的依赖性会显著降低编码的性能。为了通过图设计来控制批次依赖性，我们提出以结构化方式构造BATS码。本文提出了一种硬件友好的结构化BATS码，称为循环移位BATS（CS-BATS）码，该码通过轻量级循环移位操作从一个小型基图构造编码。我们证明，当基图设计恰当时，与随机BATS码相比，可以获得更高的解码率和更低的复杂度。