Multicast Network Coding and Field Sizes
Bibliographic record
Abstract
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 paper, 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. In particular, we prove by construction that: 1) for every ω ≥ 3, there is a multicast network with source outdegree ω linearly solvable over GF(7) but not over GF(8), and another multicast network linearly solvable over GF(16) but not over GF(17); 2) 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; 3) a multicast network linearly solvable over GF(q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m1</sup> ) and over GF(q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m2</sup> ) is not necessarily linearly solvable over GF(q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m1+m2</sup> ); and 4) there exists a class of multicast networks with a set T of receivers such that the minimum field size q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> for a linear solution over GF(q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> ) is lower bounded by O(√|T|), but not every larger field than GF(q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> ) suffices to yield a linear solution. The insight brought from this paper 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.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".