Circular-Shift Linear Network Coding

We study a class of linear network coding (LNC) schemes, called <italic>circular-shift</italic> LNC, whose encoding operations consist of only circular-shifts and bit-wise additions. Formulated as a special vector linear code over GF(2), an <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>-dimensional circular-shift linear code of degree <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> restricts its local encoding kernels to be the summation of at most <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> cyclic permutation matrices of size <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>. We show that on a general network, for a certain block length <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>, every scalar linear solution over GF(<inline-formula> <tex-math notation="LaTeX">$2^{L-1}$ </tex-math></inline-formula>) can induce an <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> can be efficiently constructed, which has an interesting complexity tradeoff between encoding and decoding with different choices of <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>, a randomly constructed circular-shift linear code has linear solvability behavior comparable to a randomly constructed permutation-based linear code, but has shorter overheads.

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