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The article presents observations about pyramidal arrangements associated with the mathematical designs of philosopher Johannes Kepler. Kepler, who first realized that planets orbit the sun in ellipses, conjectured in 1611 that fruit sellers already had it right: The best packing is the familiar pyramidal arrangement seen in markets all over the world. Despite the simplicity of this proposed solution, proving Kepler's conjecture turned out to be elusive for centuries and, in the end, required the assistance of a computer. In 1998, mathematician Thomas C. Hales made headlines by settling a nearly 400-year-old question: What is the best space-saving way to stack oranges? Hales' 250-page paper is so complex that referees have spent 6 years poring over its details, and although they still haven't checked every one, they recently gave the paper the thumbs-up to be published in digest form in the Annals of Mathematics. Hales' opus would seem to lay to rest the question of how to stack fruit. Using coding techniques, mathematicians have uncovered a surprise. Although the definitions of spheres in every dimension are analogous, the configurations of spheres that the various dimensions can contain are very different. Results from two research groups are providing new glimpses of the uniqueness of various dimensions. When it comes to three-dimensional oranges, the kissing number is 12. Musin has now proved that the kissing number in dimension four is 24. The 24 spheres that surround the central sphere trace out a highly symmetrical four-dimensional shape called the 24-cell, which has no analog in any other dimension. When the late Claude Shannon launched the theory of error-correcting codes in the 1940s, high-dimensional sphere packing instantly became a hot topic, Cohn says. "Before that, if you told someone you were interested in 24-dimensional sphere packing, unless they were a pure mathematician, they looked at you as if you were crazy," he says.

Quantum convolutional codes, like their classical counterparts, promise to offer higher error correction performance than block codes of equivalent encoding complexity, and are expected to find important applications in reliable quantum communication where a continuous stream of qubits is transmitted. Grassl and Roetteler devised an algorithm to encode a quantum convolutional code with a "pearl-necklace" encoder. Despite their algorithm's theoretical significance as a neat way of representing quantum convolutional codes, it is not well suited to practical realization. In fact, there is no straightforward way to implement any given pearl-necklace structure. This paper closes the gap between theoretical representation and practical implementation. In our previous work, we presented an efficient algorithm to find a minimal-memory realization of a pearl-necklace encoder for Calderbank-Shor-Steane (CSS) convolutional codes. This work is an extension of our previous work and presents an algorithm for turning a pearl-necklace encoder for a general (non-CSS) quantum convolutional code into a realizable quantum convolutional encoder. We show that a minimal-memory realization depends on the commutativity relations between the gate strings in the pearl-necklace encoder. We find a realization by means of a weighted graph which details the noncommutative paths through the pearl necklace. The weight of the longest path in this graph is equal to the minimal amount of memory needed to implement the encoder. The algorithm has a polynomial-time complexity in the number of gate strings in the pearl-necklace encoder. [ABSTRACT FROM AUTHOR]