Circular-Shift Linear Network Codes With Arbitrary Odd Block Lengths - Sun, Qifu Tyler; Tang, Hanqi; Li, Zongpeng; Yang, Xiaolong; Long, Keping (IEEE transactions on communications, 2019)
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 φ (L)/L , where φ (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.