Hadamard matrix conjecture

Can we make a square table using only +1 and −1, so that every pair of different rows cancels when compared? Compare rows by multiplying entries in the same positions and adding the results. For the rows (+1,+1,+1,+1) and (+1,+1,−1,−1), the result is 1 + 1 − 1 − 1 = 0. Four rows with this property can form a four-by-four table. The conjecture asks for such a table at every positive size divisible by four.[2] [3]

Can every positive multiple of four be the size of a square real sign table whose different rows cancel exactly under pairwise comparison? Multiplying corresponding row entries and summing is the dot product. At size four, use rows $r_1=(1,1,1,1)$, $r_2=(1,1,-1,-1)$, $r_3=(1,-1,1,-1)$ and $r_4=(1,-1,-1,1)$. Every distinct pair has dot product zero, while each row’s self-comparison is four.

For an $n$-by-$n$ table $H$, these comparisons form the matrix $HH^{\mathsf T}$, where the superscript means transpose. The condition is $HH^{\mathsf T}=nI_n$, with $I_n$ the matrix having ones on the diagonal and zeros elsewhere. The conjecture asks for such an $H$ with entries $\pm1$ whenever $n=4k$ and $k\ge1$. To see why four divides any order greater than two, flip columns to make the first row all positive. Orthogonality makes the second row half positive and half negative. A third row must have equally many positive and negative entries within each of those halves, forcing quarter-sized blocks and hence divisibility by four.[2] [3]

Many sizes and whole families of sizes have constructions. A construction for every positive multiple of four is missing.[2] [3]

Construction databases give explicit tables in many orders, rather than all orders. The July 2026 preprint excludes some candidates in a particular construction from a pair of periodic sign sequences, with extra common symmetries imposed. Excluding those patterned candidates does not rule out every real sign table of that order. Complex entries or additional skew conditions describe other variants.[2] [3]

Perfect cancellation lets overlapping sign patterns be separated: comparing a mixture with one row removes the contributions from all the other rows. That is why these tables are useful patterns for signal processing and statistics. The question asks which table sizes permit this separation with equally simple entries.[3]

The cancellation gives an exact recovery calculation. With the four-row example, a mixture $y=2r_1+3r_2=(5,5,-1,-1)$ has $y\cdot r_1=8$ and $y\cdot r_2=12$. Dividing by four recovers the weights two and three, because the unwanted row’s contribution cancels. This illustrates the intrinsic value of orthogonal sign patterns; it is not a performance claim about a particular device. The construction database describes applications in signal processing and statistics, where such structured comparisons are useful.[3]