Circular-shift Linear Network Codes with Arbitrary Odd Block Lengths

TL;DR

This paper extends circular-shift linear network coding to arbitrary odd block lengths, establishing conditions for solutions based on scalar linear solutions over specific finite fields, and provides an efficient construction algorithm.

Contribution

It generalizes the connection between scalar and circular-shift linear solutions for all odd lengths and introduces an efficient construction method.

Findings

Every scalar linear solution over GF(2^{m_L}) induces an L-dimensional circular-shift solution.

For large m_L, multicast networks have solutions at rate φ(L)/L.

All multicast networks are asymptotically circular-shift linearly solvable.

Abstract

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 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 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.