Sparse Rev-Shift coded modulation with novel overhead bound

To provide reliability in distributed systems, combination property (CP) is desired, where k original packets are encoded into n ≥ k packets and arbitrary k are sufficient to reconstruct all the original packets. Shift-and-add (SA) encoding combined with zigzag decoding (ZD) obtains the CP-ZD, which is promising to reap low computational complexity in the encoding/decoding process of these systems. As densely coded modulation is difficult to achieve CP-ZD, research attentions are paid to sparse coded modulation. The drawback of existing sparse CP-ZD coded modulation lies in high overhead, especially in widely deployed setting m < k, where
m\,\,\,{_=^\triangle}\,\,n-k
. For this scenario, namely, m < k, a sparse reverseorder shift (Rev-Shift) CP-ZD coded modulation is designed. The proof that Rev-Shift possesses CP-ZD is provided. A lower bound for the overhead, as far as we know is the first for sparse CP-ZD coded modulation, is derived. The bound is found tight in certain scenarios, which shows the code optimality. Extensive numerical studies show that compared to existing sparse CP-ZD coded modulation, the overhead of Rev-Shift reduces significantly, and the derived lower bound is tight when k or m approaches 0.