Circular-shift Linear Network Coding

TL;DR

This paper introduces circular-shift linear network coding, a simple scheme using only circular shifts and XOR, demonstrating its efficiency, limitations, and how it compares to other coding methods in multicast networks.

Contribution

It defines circular-shift LNC, shows how it can be constructed from scalar solutions, analyzes its optimality and limitations, and compares its performance with permutation-based codes.

Findings

Circular-shift LNC can be induced from scalar solutions over GF(2^{L-1}) with 1-bit redundancy.

Constructed circular-shift solutions are efficient and have a tradeoff between encoding complexity and decoding.

Circular-shift LNC is limited in achieving the exact network capacity, but is nearly optimal with minimal overhead.

Abstract

We study a class of linear network coding (LNC) schemes, called circular-shift LNC, whose encoding operations consist of only circular-shifts and bit-wise additions (XOR). Formulated as a special vector linear code over GF($2$), an $L$-dimensional circular-shift linear code of degree $\delta$ restricts its local encoding kernels to be the summation of at most $\delta$ cyclic permutation matrices of size $L$. We show that on a general network, for a certain block length $L$, every scalar linear solution over GF($2^{L-1}$) can induce an $L$-dimensional circular-shift linear solution with 1-bit redundancy per-edge transmission. Consequently, specific to a multicast network, such a circular-shift linear solution of an arbitrary degree $\delta$ can be efficiently constructed, which has an interesting complexity tradeoff between encoding and decoding with different choices of $\delta$. By further proving that circular-shift LNC is insufficient to achieve the exact capacity of certain multicast networks, we show the optimality of the efficiently constructed circular-shift linear solution in the sense that its 1-bit redundancy is inevitable. Finally, both theoretical and numerical analysis imply that with increasing $L$, a randomly constructed circular-shift linear code has linear solvability behavior comparable to a randomly constructed permutation-based linear code, but has shorter overheads.