More than three decades ago, mathematicians at the University of Mainz in Germany began to develop a theory that codes could be presented at a level one rung higher than sequences made up of zeros and ones: subspaces mathematics called q-analogues.
For a long time no application was found – or even sought – for the theory until a decade ago when it was understood that it would be useful in the efficient transmission of data required by modern data networks.
The challenge was that, despite many attempts, the best possible codes described in the theory had not been found and were therefore thought not to even exist.
However, an international team of mathematicians from Finland, Israel, Germany, Singapore and the United States disagreed.
“’We thought it might very well be possible. The search was difficult due to the huge size of the structures,” said team member Professor Patric Östergård, from Aalto University.
“Finding them is a gigantic operation even if there is a very high level computing capacity.”
“Therefore, in addition to algebraic techniques and computers, we also had to use our experience and guess where to start looking, and thus limit the scope of the search.”
Persistence was rewarded when Professor Östergård and his colleagues found the largest possible structure described by the theory.
The results were published online recently in the journal Mathematics Forum, Pi.
“Although mathematical breakthroughs rarely immediately become financial successes, many modern things that we take for granted would not exist without them,” the mathematicians said.
“For example, Boolean algebra, which played a key role in the creation of computers, has been developed since the 19th century,”
“In fact, information theory was green even before anyone mentioned green alternatives,” Professor Östergård added.
“Its basic idea is, in fact, to try to make the most of the power of the transmitter, which in practice means trying to transmit data using as little power as possible.”
“Our discovery will not become a product right away, but it could gradually become part of the Internet.”
Michael Brown et al. 2016. Existence of q analogues of Steiner systems. Mathematics Forum, Pi, 4: e7; doi:10.1017/fmp.2016.5
This article is based on a press release issued by Aalto University.