Circular-shift Linear Network Coding

We study a class of linear network coding (LNC) schemes, called \emph{circular-shift} LNC, whose encoding operations at intermediate nodes consist of only circular-shifts and bit-wise addition (XOR). Departing from existing literature, we systematically formulate circular-shift LNC as a special type of vector LNC, where the local encoding kernels of an $L$-dimensional circular-shift linear code of degree $\delta$ are summation of at most $\delta$ cyclic-permutation matrices of size $L$. Under this framework, an intrinsic connection between scalar LNC and circular-shift LNC is established. In particular, on a general network, for some block lengths $L$, every scalar linear solution over GF($2^{L-1}$) can induce an $(L-1, L)$-fractional circular-shift linear solution of degree $(L-1)/2$. Specific to multicast networks, an $(L-1, L)$-fractional circular-shift linear solution of arbitrary degree $\delta$ can be efficiently constructed. With different $\delta$, the constructed solution has an interesting encoding-decoding complexity tradeoff, and when $\delta = (L-1)/2$, it requires fewer binary operations for both encoding and decoding processes compared with scalar LNC. While the constructed solution has one-bit redundancy per edge transmission, we show that this is inevitable, and that circular-shift LNC is insufficient to achieve the exact capacity of some multicast networks. Last, both theoretical and numerical analysis imply that with increasing $L$, a randomly constructed circular-shift linear code has comparable linear solvability behavior to a randomly constructed permutation-based linear code, but has much shorter overheads for random coding.