  Circular-shift linear network coding (LNC) is a class of vector LNC with low
encoding and decoding complexities, and with local encoding kernels chosen from
cyclic permutation matrices. When $L$ is a prime with primitive root $2$, it
was recently shown that a scalar linear solution over GF($2^{L-1}$) induces an
$L$-dimensional circular-shift linear solution at rate $(L-1)/L$. In this work,
we prove that for arbitrary odd $L$, every scalar linear solution over
GF($2^{m_L}$), where $m_L$ refers to the multiplicative order of $2$ modulo
$L$, can induce an $L$-dimensional circular-shift linear solution at a certain
rate. Based on the generalized connection, we further prove that for such $L$
with $m_L$ beyond a threshold, every multicast network has an $L$-dimensional
circular-shift linear solution at rate $\phi(L)/L$, where $\phi(L)$ is the
Euler's totient function of $L$. An efficient algorithm for constructing such a
solution is designed. Finally, we prove that every multicast network is
asymptotically circular-shift linearly solvable.
