For each integer m ≥ 2, a network is constructed which is solvable over an alphabet of size m but is not solvable over any smaller alphabets. If m is composite, then the network has no vector linear solution over any R-module alphabet and is not asymptotically linear solvable over any finite-field alphabet. The network's capacity is shown to equal one, and when m is composite, its linear capacity is shown to be bounded away from one for all finite-field alphabets.
Fixed-size commutative rings are quasi-ordered such that all scalar linearly solvable networks over any given ring are also scalar linearly solvable over any higher-ordered ring. As consequences, if a network has a scalar linear solution over some finite commutative ring, then (i) the network is also scalar linearly solvable over a maximal commutative ring of the same size, and (ii) the (unique) smallest size commutative ring over which the network has a scalar linear solution is a field. We prove that a commutative ring is maximal with respect to the quasi-order if and only if some network is scalar linearly solvable over the ring but not over any other commutative ring of the same size. Furthermore, we show that maximal commutative rings are direct products of certain fields specified by the integer partitions of the prime factor multiplicities of the maximal ring's size.Finally, we prove that there is a unique maximal commutative ring of size m if and only if each prime factor of m has multiplicity in {1, 2, 3, 4, 6}. In fact, whenever p is prime and k ∈ {1, 2, 3, 4, 6}, the unique such maximal ring of size p k is the field GF p k . However, for every field GF p k with k ∈ {1, 2, 3, 4, 6}, there is always some network that is not scalar linearly solvable over the field but is scalar linearly solvable over a commutative ring of the same size. These results imply that for scalar linear network coding over commutative rings, fields can always be used when the alphabet size is flexible, but alternative rings may be needed when the alphabet size is fixed.
We prove the following results regarding the linear solvability of networks over various alphabets. For any network, the following are equivalent: (i) vector linear solvability over some finite field, (ii) scalar linear solvability over some ring, (iii) linear solvability over some module. Analogously, the following are equivalent: (a) scalar linear solvability over some finite field, (b) scalar linear solvability over some commutative ring, (c) linear solvability over some module whose ring is commutative. Whenever any network is linearly solvable over a module, a smallest such module arises in a vector linear solution for that network over a field.If a network is linearly solvable over some non-commutative ring but not over any commutative ring, then such a non-commutative ring must have size at least 16, and for some networks, this bound is achieved. An infinite family of networks is demonstrated, each of which is scalar linearly solvable over some non-commutative ring but not over any commutative ring.Whenever p is prime and 2 ≤ k ≤ 6, if a network is scalar linearly solvable over some ring of size p k , then it is also k-dimensional vector linearly solvable over the field GF(p), but the converse does not necessarily hold. This result is extended to all k ≥ 2 when the ring is commutative.
RF-modulated electron beams, such as those produced by an RF linear accelerator, propagating through vacuum, air, and solid matter are well known to drive signals in microwave cavities and waveguides via interactions with these structures. Past experiments with a microwave waveguide in a radiation-shielded vault indicated the presence of a multipath propagation phenomenon, hypothesized to be a result of reflections of RF-modulated x rays. In this work, we study the signals induced in a microwave coaxial cable from nearby beam interactions with materials commonly found in accelerator facilities in order to better understand RF production and propagation in these environments. Our results show that (1) when an RF-modulated electron beam is incident on a block of aluminum, lead, or concrete, the frequency content of the induced microwave signals is strongly dependent on the orientation of the block and the relative position of the detector, (2) at least some of the detected signals are consistent with reflections off of the blocks, and (3) beam interactions with the blocks can induce appreciable microwave signals in detectors located tens of cm from the block.
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