Circular-Shift Linear Network Codes With Arbitrary Odd Block Lengths

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 mL ), 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 φ(L)/L, where φ(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. DRAFTJanuary 3, 2019 1 Throughout this paper, the notion [Ae] e∈E ′ will always refer to column-wise juxtaposition of matrices Ae with e orderly chosen from a subset E ′ of E, where Ae may degenerate to vectors. DRAFT (26)For simplicity, writeTogether with (25), Eq. (26) can be written as U 1Ṽ + U 2 = I J . Because U T 1 andṼ can be respectively regarded as a Vandermonde matrix generated by α −j , j ∈ J and by α j , j ∈ J , they are invertible, and so is I J + U 2 . Thus, C. Proof of Theorem 4It remains to prove (11). Follow the same argument as in the proof of Theorem 1 (refer to Appendix-A) till Eq. (24). For a receiver t, by (21), (22) and (24), we have [F e ] e∈In(t) = (I ω ⊗ V L )P TM P(I ω ⊗ V −1 L ), whereM = M 0 0 0 0 . . . 0 0 0 M L−1 , and M j = [f e (α j )] e∈In(t) .