On Vector Random Linear Network Coding in Wireless Broadcasts

Compared with scalar linear network coding (LNC) formulated over the finite field GF(2L), vector LNC offers enhanced flexibility in the code design by enabling linear operations over the vector space GF(2)L and demonstrates a number of advantages over scalar LNC. While random LNC (RLNC) has shown significant potential to improve the completion delay performance in wireless broadcasts, most prior studies focus on scalar RLNC. In particular, it is well known that, with increasing L, primitive scalar RLNC over GF(2L) asymptotically achieves the optimal completion delay. However, the completion delay performance of primitive vector RLNC remains unexplored. This work aims to fill in this blank. We derive closed-form expressions for the probability distribution and the expected value of both the completion delay at a single receiver and the system completion delay. We further unveil a fundamental limitation that is different from scalar RLNC: even for large enough L, primitive vector RLNC over GF(2)L inherently fails to reach optimal completion delay. In spite of this, the gap between the expected completion delay at a receiver and the optimal one is shown to be a constant smaller than 0.714, which implies that the expected completion delay normalized by the number P of original packets is asymptotically optimal with increasing P. We also validate our theoretical characterization through numerical simulations. Our theoretical characterization establishes primitive vector RLNC as a performance baseline for the future design of practical vector RLNC schemes with different design goals.