We study a class of linear network coding (LNC) schemes, called circular-shift LNC, whose encoding operations consist of only circular-shifts and bit-wise additions (XOR). Formulated as a special vector linear code over GF(2), an L-dimensional circular-shift linear code of degree δ restricts its local encoding kernels to be the summation of at most δ cyclic permutation matrices of size L. We show that on a general network, for a certain block length L, every scalar linear solution over GF(2 L−1 ) can induce an L-dimensional circular-shift linear solution with 1-bit redundancy per-edge transmission.Consequently, specific to a multicast network, such a circular-shift linear solution of an arbitrary degree δ can be efficiently constructed, which has an interesting complexity tradeoff between encoding and decoding with different choices of δ. By further proving that circular-shift LNC is insufficient to achieve the exact capacity of certain multicast networks, we show the optimality of the efficiently constructed circular-shift linear solution in the sense that its 1-bit redundancy is inevitable. Finally, both theoretical and numerical analysis imply that with increasing L, a randomly constructed circular-shift linear code has linear solvability behavior comparable to a randomly constructed permutation-based linear code, but has shorter overheads.

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In an acyclic multicast network, it is well known that a linear network coding solution over GF($q$) exists when $q$ is sufficiently large. In particular, for each prime power $q$ no smaller than the number of receivers, a linear solution over GF($q$) can be efficiently constructed. In this work, we reveal that a linear solution over a given finite field does \emph{not} necessarily imply the existence of a linear solution over all larger finite fields. Specifically, we prove by construction that: (i) For every source dimension no smaller than 3, there is a multicast network linearly solvable over GF(7) but not over GF(8), and another multicast network linearly solvable over GF(16) but not over GF(17); (ii) There is a multicast network linearly solvable over GF(5) but not over such GF($q$) that $q > 5$ is a Mersenne prime plus 1, which can be extremely large; (iii) A multicast network linearly solvable over GF($q^{m_1}$) and over GF($q^{m_2}$) is \emph{not} necessarily linearly solvable over GF($q^{m_1+m_2}$); (iv) There exists a class of multicast networks with a set $T$ of receivers such that the minimum field size $q_{min}$ for a linear solution over GF($q_{min}$) is lower bounded by $\Theta(\sqrt{|T|})$, but not every larger field than GF($q_{min}$) suffices to yield a linear solution. The insight brought from this work is that not only the field size, but also the order of subgroups in the multiplicative group of a finite field affects the linear solvability of a multicast network

In an acyclic multicast network, it is well known that a linear network coding solution over GF(q) exists when q is sufficiently large. In particular, for each prime power q no smaller than the number of receivers, a linear solution over GF(q) can be efficiently constructed. In this work, we reveal that a linear solution over a given finite field does not necessarily imply the existence of a linear solution over all larger finite fields. Specifically, we prove by construction that: (i) For every source dimension no smaller than 3, there is a multicast network linearly solvable over GF(7) but not over GF(8), and another multicast network linearly solvable over GF(16) but not over GF(17); (ii) There is a multicast network linearly solvable over GF(5) but not over such GF(q) that q > 5 is a Mersenne prime plus 1, which can be extremely large; (iii) A multicast network linearly solvable over GF(q m1 ) and over GF(q m2 ) is not necessarily linearly solvable over GF(q m1+m2 ); (iv) There exists a class of multicast networks with a set T of receivers such that the minimum field size q min for a linear solution over GF(q min ) is lower bounded by Θ( |T |), but not every larger field than GF(q min ) suffices to yield a linear solution. The insight brought from this work is that not only the field size, but also the order of subgroups in the multiplicative group of a finite field affects the linear solvability of a multicast network. Index TermsLinear network coding, multicast network, field size, lower bound, Mersenne prime.

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In the literature of network coding, vector linear network coding (LNC) is a generalization of the conventional scalar LNC, such that the data unit transmitted on every edge is an L-dimensional vector of data symbols over a base field GF(q). A scalar linear code over GF(q) is simply a vector linear code of dimension 1 over GF(q), and a general network has a scalar linear solution over GF(q L ) only if it has a vector linear solution of dimension L over GF(q). Though vector LNC is more powerful in enabling a higher coding diversity, this work will present explicit multicast networks, for the first time in the literature, with the special property that they do not have a vector linear solution of dimension L over GF(2) but have scalar linear solutions over GF(q ′ ), for some q ′ < 2 L . This reveals the fact that although vector LNC can outperform scalar LNC in terms of yielding a solution for a general network, scalar LNC can also outperform vector LNC of dimension larger than 1 in terms of using a smaller alphabet to yield a solution for a multicast network. IEEE ICC 2015 -Communication Theory Symposium978-1-4673-6432-4/15/$31.00 ©2015 IEEE

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In this paper, a time-variant decoding model of a convolutional network code (CNC) is proposed. New necessary and sufficient conditions are established for the decodability of a CNC at a node r with delay L. They only involve the first L + 1 terms in the power series expansion of the global encoding kernel matrix at r. Concomitantly, a time-variant decoding algorithm is proposed with a decoding matrix over the base symbol field. The present time-variant decoding model only deals with partial information of the global encoding kernel matrix, and hence potentially makes CNCs applicable in a decentralized manner.