Editing Hadamard matrix (section)

==Equivalence and uniqueness==
Two Hadamard matrices are considered [[equivalence relation|equivalent]] if one can be obtained from the other by negating rows or columns, or by interchanging rows or columns. Up to equivalence, there is a unique Hadamard matrix of orders 1, 2, 4, 8, and 12. There are 5 inequivalent matrices of order 16, 3 of order 20, 60 of order 24, and 487 of order 28. Millions of inequivalent matrices are known for orders 32, 36, and 40. Using a [[equivalence relation#Comparing equivalence relations|coarser]] notion of equivalence that also allows [[transpose|transposition]], there are 4 inequivalent matrices of order 16, 3 of order 20, 36 of order 24, and 294 of order 28.<ref>{{cite journal|last=Wanless|first=I.M.|title=Permanents of matrices of signed ones|journal=Linear and Multilinear Algebra |year=2005 |volume=53 |issue=6|pages=427–433 |doi=10.1080/03081080500093990|s2cid=121547091}}</ref>

Hadamard matrices are also uniquely recoverable, in the following sense: If an Hadamard matrix  <math>H</math> of order <math>n</math> has <math>O(n^2/\log n)</math> entries randomly deleted, then with overwhelming likelihood, one can perfectly recover the original matrix <math>H</math> from the damaged one. The algorithm of recovery has the same computational cost as matrix inversion.<ref>{{cite journal|last=Kline|first=J.|title=Geometric search for Hadamard matrices|journal=Theoretical Computer Science|year=2019 |volume=778 |pages=33–46|doi=10.1016/j.tcs.2019.01.025|s2cid=126730552|doi-access=free}}</ref>