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, 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 paper, 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. [ABSTRACT FROM AUTHOR]