The best possible q-analogs of codes may be useful in more efficient data transmission. Image credit: Geralt.
More than three decades ago, mathematicians at the University of Mainz in Germany started developing a theory according to which codes could be presented at a level one step higher than the sequences formed by zeros and ones: mathematical subspaces named q-analogs.
For a long time, no applications were found — or were not even searched for — for the theory until a decade ago, when it was understood that they would be useful in the efficient data transmission required by modern data networks.
The challenge was that, despite numerous attempts, the best possible codes described in the theory had not been found and it was therefore believed they did not even exist.
However, an international team of mathematicians from Finland, Israel, Germany, Singapore and the United States disagreed.
“’We thought it could very well be possible. The search was challenging because of the enormous size of the structures,” said team member Prof. Patric Östergård, of Aalto University.
“Searching for them is a gigantic operation even if there is very high-level computational capacity available.”
“Therefore, in addition to algebraic techniques and computers, we also had to use our experience and guess where to start looking, and that way limit the scope of the search.”
The perseverance was rewarded when Prof. Östergård and his colleagues found the largest possible structure described by the theory.
The results were published online recently in the journal Forum of Mathematics, Pi.
“Although mathematical breakthroughs rarely become financial success stories immediately, many modern things we take for granted would not exist without them,” the mathematicians said.
“For example, Boolean algebra, which has played a key role in the creation of computers, has been developed since the 19th century,”
“As a matter of fact, information theory was green before anyone had even mentioned green alternatives,” Prof. Östergård added.
“Its basic idea is, actually, to try to take advantage of the power of the transmitter as effectively as possible, which in practice means attempting to transmit data using as little energy as possible.”
“Our discovery will not become a product straight away, but it may gradually become part of the Internet.”
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Michael Braun et al. 2016. Existence of q-analogs of Steiner systems. Forum of Mathematics, Pi, 4: e7; doi: 10.1017/fmp.2016.5
This article is based on a press-release issued by Aalto University.
