Circular-Shift Linear Network Codes With Arbitrary Odd Block Lengths

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 <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> is a prime with primitive root 2, it was recently shown that a scalar linear solution over GF(<inline-formula> <tex-math notation="LaTeX">$2^{L-1}$ </tex-math></inline-formula>) induces an <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>-dimensional circular-shift linear solution at rate <inline-formula> <tex-math notation="LaTeX">$(L-1)/L$ </tex-math></inline-formula>. In this paper, we prove that for arbitrary odd <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>, every scalar linear solution over GF(<inline-formula> <tex-math notation="LaTeX">$2^{m_{L}}$ </tex-math></inline-formula>), where <inline-formula> <tex-math notation="LaTeX">$m_{L}$ </tex-math></inline-formula> refers to the multiplicative order of 2 modulo <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>, can induce an <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>-dimensional circular-shift linear solution at a certain rate. Based on the generalized connection, we further prove that for such <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$m_{L}$ </tex-math></inline-formula> beyond a threshold, every multicast network has an <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>-dimensional circular-shift linear solution at rate <inline-formula> <tex-math notation="LaTeX">$\phi (L)/L$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\phi (L)$ </tex-math></inline-formula> is the Euler’s totient function of <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>. An efficient algorithm for constructing such a solution is designed. Finally, we prove that every multicast network is asymptotically circular-shift linearly solvable.

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