  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 (XOR). Formulated as a special vector linear code over
GF($2$), an $L$-dimensional circular-shift linear code of degree $\delta$
restricts its local encoding kernels to be the summation of at most $\delta$
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 $\delta$ can be
efficiently constructed, which has an interesting complexity tradeoff between
encoding and decoding with different choices of $\delta$. 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.
