Circular-Shift Linear Network Codes With Arbitrary Odd Block Lengths

• Published in: IEEE transactions on communications Vol. 67; no. 4; pp. 2660 - 2672
• Main Authors: Sun, Qifu Tyler, Tang, Hanqi, Li, Zongpeng, Yang, Xiaolong, Long, Keping
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.04.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; circular-shift; Circularity; Coding; Decoding; efficient construction; fractional code; Kernel; Linear codes; Multicast; Network coding; Permutations; Receivers; Sun; vector linear code
• ISSN: 0090-6778; 1558-0857
• DOI: 10.1109/TCOMM.2018.2890260

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.