Circular-shift linear network coding

Abstract

We study a class of linear network coding (LNC) schemes, called circular-shift LNC, whose encoding operations at intermediate nodes consist of only circular-shifts and bitwise addition (XOR). Departing from existing literature, we systematically formulate circular-shift LNC as a special type of vector LNC, where the local encoding kernels of an L-dimensional circular-shift linear code of degree δ are summation of at most δ cyclic-permutation matrices of size L. Under this framework, an intrinsic connection between scalar LNC and circular-shift LNC is established. In consequence, for some block lengths L, an (L − 1, L)-fractional circular-shift linear solution of arbitrary degree δ can be efficiently constructed on a multicast network. With different δ, the constructed solution has an interesting encoding-decoding complexity tradeoff, and when δ = (L − 1)/2, it requires fewer binary operations for both encoding and decoding processes compared with scalar LNC. While the constructed (L − 1, L)-fractional solution has one-bit redundancy per edge transmission, we show that this is inevitable, and that circular-shift LNC is insufficient to achieve the exact capacity of multicast networks.