Circular-Shift Linear Network Coding - Tang, Hanqi; Sun, Qifu Tyler; Li, Zongpeng; Yang, Xiaolong; Long, Keping (IEEE transactions on information theory, 2019)
We study a class of linear network coding (LNC) schemes, called circular-shift LNC, whose encoding operations consist of only circular-shifts and bit-wise additions. Formulated as a special vector linear code over GF(2), an L -dimensional circular-shift linear code of degree δ restricts its local encoding kernels to be the summation of at most δ cyclic permutation matrices of size L . We show that on a general network, for a certain block length L , every scalar linear solution over GF( 2^{L-1} ) can induce an L -dimensional circular-shift linear solution with 1-bit redundancy per-edge transmission. Consequently, specific to a multicast network, such a circular-shift linear solution of an arbitrary degree δ can be efficiently constructed, which has an interesting complexity tradeoff between encoding and decoding with different choices of δ . By further proving that circular-shift LNC is insufficient to achieve the exact capacity of certain multicast networks, we show the optimality of the efficiently constructed circular-shift linear solution in the sense that its 1-bit redundancy is inevitable. Finally, both theoretical and numerical analysis imply that with increasing L , a randomly constructed circular-shift linear code has linear solvability behavior comparable to a randomly constructed permutation-based linear code, but has shorter overheads.