On Base , Normal and Near-normal Sequences

In the quiet, abstract world of combinatorial mathematics, researchers often hunt for patterns hidden within simple lists of numbers. Imagine a long row of switches, each set to either "on" or "off." If you take two such rows and compare them, sliding one past the other, you can measure how much they match or clash at every step. This measurement is called an autocorrelation. For decades, mathematicians have been fascinated by a specific puzzle: can you find four of these rows of switches, arranged in two pairs of different lengths, such that when you slide them around and add up all their matches and clashes, the total noise cancels out perfectly to zero everywhere except at the very start? This is the quest for "base sequences."

These sequences are not just a game of numbers; they are the hidden blueprints for building a special kind of square grid known as a Hadamard matrix. These grids are vital tools in engineering and physics, used to create signals that can travel through noisy environments without getting garbled, much like how a clear voice cuts through a crowded room. The bigger the grid, the more powerful the signal, but finding the right arrangement of switches to build these grids becomes exponentially harder as the size increases. For years, mathematicians had a strong hunch, called a conjecture, that these perfect arrangements could be found for every possible size, but they could only prove it for sizes up to a certain point.

A team of researchers at Nanjing University of Information Science and Technology has now pushed the boundaries of this search further than ever before. They developed a new, highly efficient method to sift through the billions of possible combinations of these switch rows. Instead of trying to guess all four rows at once, their algorithm first builds pairs of rows that fit together, and then searches for the remaining two to complete the set. Using this approach, they successfully constructed these perfect sequences for sizes 41, 42, and 43 for the first time in history. This achievement confirms that the long-standing guess about the existence of these sequences holds true for all sizes up to 43, extending the known territory of these mathematical structures.

However, the story takes a surprising turn when the researchers looked at two specific, stricter versions of these sequences, known as "near-normal" sequences. For these, there was a similar belief that they could be found for every even number. The team ran an exhaustive computer search to check this for the next few even numbers. They found that while these sequences exist for many sizes, they simply do not exist for the numbers 42 and 44. This discovery is significant because it provides the first concrete proof that the belief about near-normal sequences is false. It is a rare moment in mathematics where a long-held idea is not just unproven, but explicitly shown to be wrong by finding a specific case where it fails.

The researchers also investigated another strict version called "normal" sequences. They proved mathematically that these sequences cannot exist for a specific pattern of numbers, specifically those that are two less than a multiple of eight. Their computer search further confirmed that these sequences do not exist for the numbers 41 through 46. While they have successfully built the general sequences for the new sizes, the existence of the stricter versions for larger numbers remains a mystery. The work does not just add new examples to a list; it reshapes the landscape of what is possible, showing exactly where the patterns break down and where they continue to hold.

The implications of these findings ripple outward to the construction of the Hadamard matrices mentioned at the start. Since these matrices are essential for creating robust communication systems and error-correcting codes, knowing exactly which sizes can be built and which cannot is crucial. The researchers' new algorithm, which combines deep mathematical rules with powerful computer searching, offers a clear path forward. It allows scientists to test even larger sizes with confidence, knowing that the method can filter out impossible combinations quickly. While the question of whether these sequences exist for every single size is now known to be false for the stricter types, the general quest continues, with the researchers having cleared the path for the next generation of discoveries in this intricate field.