Circular-Shift Linear Network Codes With Arbitrary Odd Block Lengths
Qifu Tyler Sun, Hanqi Tang, Zongpeng Li, Xiaolong Yang, Keping Long · IEEE Transactions on Communications · 2018
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(2L-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(2mL), where mL 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 mL 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.