Circularshift Linear Network Codes with Arbitrary Odd Block Lengths
Abstract
Circularshift 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($2^{L1}$) induces an $L$dimensional circularshift linear solution at rate $(L1)/L$. In this work, 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 circularshift 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 circularshift 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 circularshift linearly solvable.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.04635
 Bibcode:
 2018arXiv180604635S
 Keywords:

 Computer Science  Information Theory
 EPrint:
 doi:10.1109/TCOMM.2018.2890260