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Circular-shift linear network coding (LNC) is a class of vector LNC with lowencoding and decoding complexities, and with local encoding kernels chosen fromcyclic permutation matrices. When $L$ is a prime with primitive root $2$, itwas 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 overGF($2^{m_L}$), where $m_L$ refers to the multiplicative order of $2$ modulo$L$, can induce an $L$-dimensional circular-shift linear solution at a certainrate. Based on the generalized connection, we further prove that for such $L$with $m_L$ beyond a threshold, every multicast network has an $L$-dimensionalcircular-shift linear solution at rate $\phi(L)/L$, where $\phi(L)$ is theEuler's totient function of $L$. An efficient algorithm for constructing such asolution is designed. Finally, we prove that every multicast network isasymptotically circular-shift linearly solvable.

Circular-shift Linear Network Codes with Arbitrary Odd Block Lengths