Three new lengths for cyclic Legendre pairs

Abstract

There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v = 77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing the number of undecided cases to 17.

Legendre pair of length $77$ using complementary binary matrices with fixed marginals

“…. , n}, each vector generated through simultaneous decompression of all q i and p i respectively will satisfy n j=1 (δ j − 1) of the constraints defined by equation (2). This is the case for pairs q 1 , p 1 and q 2 , p 2 with PSD constant γ = 18 = (35 + 1)/2.…”

“…By Theorem 2.3, the search space can be restricted to decimation class representatives for each δ i -compression, q i . The case n = 2 is of particular interest as next five unsolved cases for LP existence are of length {77, 85, 87, 115, 117}, each the product of exactly two prime powers [2]. Let θ : {0, 1} → {0, 1} δ 1 ×δ 2 be the mapping of binary vectors of length to arrays of size δ 1 × δ 2 , with indices mapped as defined by equation ( 9).…”

“…A total of 18, 042 inequivalent LP were identified spanning 36, 050 decimation classes. Known solutions were verified (up to decimation class equivalence) to be in the solution set [2,4,10].…”

“…There exists a construction for Hadamard matrices with size n = 2 + 2 based on LP. Just as Hadamard matrices are conjectured to exist for all n divisible by 4, LP are conjectured to exist for all odd [2]. The smallest unresolved case for LP has long been = 77 [7].…”

“…This is the case for the remaining unsolved cases for LP existence with < 200, excluding 169 = 13 2 . The cases for ∈ {77, 85, 87, 115, 117, 129, 133, 145, 147, 159, 161, 169, 175, 177, 185, 187, 195} constitute the current list of unsolved cases [2]. The result of this expansion are pairs of binary matrices with fixed row and column sums, also known as binary matrices with fixed marginals [3].…”