Circular-shift Linear Network Codes with Arbitrary Odd Block Lengths

Circular-shift linear network coding (LNC) is a class of vector LNC with low encoding and decoding complexities, 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 an arbitrary odd L, every scalar linear solution over $\mathrm{G}\mathrm{F}(2^{m_{L}})$, 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 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 and $m_{L}$ is beyond a threshold. Stemming from this, we last prove that every multicast network is asymptotically circular-shift linearly solvable.