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Circular-Shift Linear Network Codes With Arbitrary Odd Block Lengths
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DOI：10.1109/TCOMM.2018.2890260.png)
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Abstract
En 中文
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 paper, 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 phi(L)/L, where phi(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.
Keywords:
Network coding
circular-shift
vector linear code
fractional code
efficient construction
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