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Keywords: Hadamard matrices, Kimura matrices Paley matrices, Plackett-Burman designs, Sylvester matrices, Turyn construction, Williamson construction

• Contains all Hadamard matrices of orders n up through 28, and at least one of every order n up through 256.

• This library is maintained by N. J. A. Sloane. Email address: njasloane@gmail.com

• Notation:
  • had.n.name indicates a Hadamard matrix of order n and type "name".
  • The matrices are usually given as n rows each containing n +'s and -'s (with no spaces).
  • In many cases there are further rows giving the name of the matrix and the order of its automorphism group.

• What the suffixes mean:
  
  • od = orthogonal design construction method
  • pal = first Paley type
  • pal2 = second Paley type
  • syl = Sylvester type
  • tur = Turyn type
  • tx = tensor product of type x with ++/+- or (rarely) with a Hadamard matrix of order 4
  • will = Williamson type

• References:
  • Seberry, J. and Yamada, M., Hadamard matrices, sequences, and block designs, pp. 431-560 of Dinitz, J. H. and Stinson, D. R., editors (1992), Contemporary Design Theory: A Collection of Essays, Wiley, New York.
  • Chapter 7 of Orthogonal Arrays by Hedayat, Sloane and Stufken.
  • D. P. Gerakoulis and S. S. Ghassemzadeh, System and method for generating orthogonal codes, U.S. patent 6,700,864 [Application to wireless communications]
• Links:
  • Orthogonal Arrays:
    (a) Parameters of orthogonal arrays with up to 100 runs
    (b) Library of explicit orthogonal arrays
    (c) The othogonal array book.
  • Jennifer Seberry's home page contains among other things libraries of
    • 26 Hadamard matrices of order 32
    • 190 Hadamard matrices of order 36
    • Williamson matrices
    • "good" matrices
    • weighing matrices and sequences
    • SBIBD's
    • D-optimal designs
    • "almost good" matrices
    • and "profiles" of the Hadamard matrices of order 24

• Comments or contributions of further matrices are welcomed.
  • Thanks to my colleague Saeed S. Ghassemzadeh (saeedg@att.com) for supplying examples of matrices of orders 204 through 252.
  • Thanks to Hadi Kharaghani and Behruz Tayfeh-Rezaie for their newly-discovered Hadamard matrix of order 428.

• The number of inequivalent Hadamard matrices of order n is known only for n <= 32. The numbers of orders
  0, 4, 8, 12, 16, 20, 24, 28, 32,
  

  are respectively
  

  1, 1, 1, 1, 5, 3, 60, 487, 13710027
  

  -- see Sequence A007299 of the On-Line Encyclopedia of Integer Sequences.

  The total number of orbits on columns has also been calculated for orders up through 32 (this gives the number of nonisomorphic OA(n,n-1,2,2)'s) -- see Sequence A048885.

• Small orders.
  • The matrices of orders 1, 2, 4, 8 and 12 are unique (up to isomorphism) and can be taken to be
    • had.1
      
    • had.2
      
    • had.4
      
    • had.8
      
    • had.12
      

  • The five distinct Hadamard matrices of order 16 can be taken to be:
    
    • had.16.0
    • had.16.1
    • had.16.2
    • had.16.3
    • had.16.4
    These 5 matrices have been verified to be inequivalent. (The other three matrices of order 16 listed below are equivalent to these.)

  • The three distinct Hadamard matrices of order 20 can be taken to be:
    
    • had.20.pal
    • had.20.will ( isomorphic to had.20.toncheviii and to had.20.julin )
    • had.20.toncheviv
    These 3 matrices have been verified to be inequivalent.

  • The 60 matrices of order 24 in files Had.24.1 through Had.24.60 (see below) have been verified to be inequivalent.

  • The 487 matrices of order 28 in files Had.28.1 through Had.28.487 (see below) have been verified to be inequivalent.

• Complete Listing.

  At present the table contains the following Hadamard matrices:
  had.1
  had.2
  had.4
  had.8
  had.8.1
  had.12
  had.16.0
  had.16.hed
  had.16.syl
  had.16.twin
  had.16.1
  had.16.2
  had.16.3
  had.16.4
  had.20.hall.n
  had.20.julin
  had.20.pal
  had.20.toncheviii
  had.20.toncheviv
  had.20.will
  had.20.1
  had.20.2
  had.20.3
  had.24.pal
  had.24.1
  had.24.2
  had.24.3
  had.24.4
  had.24.5
  had.24.6
  had.24.7
  had.24.8
  had.24.9
  had.24.10
  had.24.11
  had.24.12
  had.24.13
  had.24.14
  had.24.15
  had.24.16
  had.24.17
  had.24.18
  had.24.19
  had.24.20
  had.24.21
  had.24.22
  had.24.23
  had.24.24
  had.24.25
  had.24.26
  had.24.27
  had.24.28
  had.24.29
  had.24.30
  had.24.31
  had.24.32
  had.24.33
  had.24.34
  had.24.35
  had.24.36
  had.24.37
  had.24.38
  had.24.39
  had.24.40
  had.24.41
  had.24.42
  had.24.43
  had.24.44
  had.24.45
  had.24.46
  had.24.47
  had.24.48
  had.24.49
  had.24.50
  had.24.51
  had.24.52
  had.24.53
  had.24.54
  had.24.55
  had.24.56
  had.24.57
  had.24.58
  had.24.59
  had.24.60
  had.28.pal2
  had.28.will
  had.32.pal
  had.32.syl
  had.32.t1
  had.32.t2
  had.32.t3
  had.32.t4
  had.36.pal2
  had.36.will
  had.40.tpal
  had.40.ttoncheviv
  had.40.twill
  had.44.pal
  had.44.will
  had.48.pal
  had.52.will
  had.56.tpal2
  had.56.twill
  had.60.pal
  had.60.pal2
  had.60.will
  had.64.syl
  had.68.pal
  had.68.will
  had.72.pal
  had.76.pal2
  had.76.will
  had.80.pal
  had.84.pal
  had.84.pal2
  had.84.will
  had.88.tpal
  had.88.twill
  had.92.will
  had.96.tpal
  had.100.will
  had.104.pal
  had.108.pal
  had.108.pal2
  had.108.will
  had.112.ttpal2
  had.112.ttwill
  had.116.will
  had.120.tpal
  had.120.tpal2
  had.120.twill
  had.124.pal2
  had.124.will
  had.128.syl
  had.132.pal
  had.132.will
  had.136.tpal
  had.136.twill
  had.140.pal
  had.144.tpal
  had.148.pal2
  had.148.will
  had.152.pal
  had.156.will
  had.160.tpal
  had.164.pal
  had.168.pal
  had.172.will
  had.176.ttpal
  had.176.ttwill
  had.180.pal
  had.184.twill
  had.188.tur
  had.192.pal
  had.196.pal2
  had.200.pal
  had.204.pal2
  had.208.twill
  had.212.pal
  had.216.tpal
  had.220.pal2
  had.224.pal
  had.228.pal
  had.232.twill
  had.236.od
  had.240.pal
  had.244.will
  had.248.twill
  had.252.pal
  had.256.syl,
  

  Added Jun 21 2004: Hadi Kharaghani and Behruz Tayfeh-Rezaie have announced the discovery of a Hadamard matrix of order 428: see this link With their kind permission, this matrix is included here.

• Also included here is Kimura's complete list of the 487 Hadamard matrices of order 28:

  Had28.tar.gz, a gzipped tar file of the 487 matrices

  The individual matrices:

  had.28.1 had.28.1 had.28.2 had.28.3 had.28.4 had.28.5 had.28.6 had.28.7 had.28.8 had.28.9 had.28.10 had.28.11 had.28.12 had.28.13 had.28.14 had.28.15 had.28.16 had.28.17 had.28.18 had.28.19 had.28.20 had.28.21 had.28.22 had.28.23 had.28.24 had.28.25 had.28.26 had.28.27 had.28.28 had.28.29 had.28.30 had.28.31 had.28.32 had.28.33 had.28.34 had.28.35 had.28.36 had.28.37 had.28.38 had.28.39 had.28.40 had.28.41 had.28.42 had.28.43 had.28.44 had.28.45 had.28.46 had.28.47 had.28.48 had.28.49 had.28.50 had.28.51 had.28.52 had.28.53 had.28.54 had.28.55 had.28.56 had.28.57 had.28.58 had.28.59 had.28.60 had.28.61 had.28.62 had.28.63 had.28.64 had.28.65 had.28.66 had.28.67 had.28.68 had.28.69 had.28.70 had.28.71 had.28.72 had.28.73 had.28.74 had.28.75 had.28.76 had.28.77 had.28.78 had.28.79 had.28.80 had.28.81 had.28.82 had.28.83 had.28.84 had.28.85 had.28.86 had.28.87 had.28.88 had.28.89 had.28.90 had.28.91 had.28.92 had.28.93 had.28.94 had.28.95 had.28.96 had.28.97 had.28.98 had.28.99 had.28.100 had.28.101 had.28.102 had.28.103 had.28.104 had.28.105 had.28.106 had.28.107 had.28.108 had.28.109 had.28.110 had.28.111 had.28.112 had.28.113 had.28.114 had.28.115 had.28.116 had.28.117 had.28.118 had.28.119 had.28.120 had.28.121 had.28.122 had.28.123 had.28.124 had.28.125 had.28.126 had.28.127 had.28.128 had.28.129 had.28.130 had.28.131 had.28.132 had.28.133 had.28.134 had.28.135 had.28.136 had.28.137 had.28.138 had.28.139 had.28.140 had.28.141 had.28.142 had.28.143 had.28.144 had.28.145 had.28.146 had.28.147 had.28.148 had.28.149 had.28.150 had.28.151 had.28.152 had.28.153 had.28.154 had.28.155 had.28.156 had.28.157 had.28.158 had.28.159 had.28.160 had.28.161 had.28.162 had.28.163 had.28.164 had.28.165 had.28.166 had.28.167 had.28.168 had.28.169 had.28.170 had.28.171 had.28.172 had.28.173 had.28.174 had.28.175 had.28.176 had.28.177 had.28.178 had.28.179 had.28.180 had.28.181 had.28.182 had.28.183 had.28.184 had.28.185 had.28.186 had.28.187 had.28.188 had.28.189 had.28.190 had.28.191 had.28.192 had.28.193 had.28.194 had.28.195 had.28.196 had.28.197 had.28.198 had.28.199 had.28.200 had.28.201 had.28.202 had.28.203 had.28.204 had.28.205 had.28.206 had.28.207 had.28.208 had.28.209 had.28.210 had.28.211 had.28.212 had.28.213 had.28.214 had.28.215 had.28.216 had.28.217 had.28.218 had.28.219 had.28.220 had.28.221 had.28.222 had.28.223 had.28.224 had.28.225 had.28.226 had.28.227 had.28.228 had.28.229 had.28.230 had.28.231 had.28.232 had.28.233 had.28.234 had.28.235 had.28.236 had.28.237 had.28.238 had.28.239 had.28.240 had.28.241 had.28.242 had.28.243 had.28.244 had.28.245 had.28.246 had.28.247 had.28.248 had.28.249 had.28.250 had.28.251 had.28.252 had.28.253 had.28.254 had.28.255 had.28.256 had.28.257 had.28.258 had.28.259 had.28.260 had.28.261 had.28.262 had.28.263 had.28.264 had.28.265 had.28.266 had.28.267 had.28.268 had.28.269 had.28.270 had.28.271 had.28.272 had.28.273 had.28.274 had.28.275 had.28.276 had.28.277 had.28.278 had.28.279 had.28.280 had.28.281 had.28.282 had.28.283 had.28.284 had.28.285 had.28.286 had.28.287 had.28.288 had.28.289 had.28.290 had.28.291 had.28.292 had.28.293 had.28.294 had.28.295 had.28.296 had.28.297 had.28.298 had.28.299 had.28.300 had.28.301 had.28.302 had.28.303 had.28.304 had.28.305 had.28.306 had.28.307 had.28.308 had.28.309 had.28.310 had.28.311 had.28.312 had.28.313 had.28.314 had.28.315 had.28.316 had.28.317 had.28.318 had.28.319 had.28.320 had.28.321 had.28.322 had.28.323 had.28.324 had.28.325 had.28.326 had.28.327 had.28.328 had.28.329 had.28.330 had.28.331 had.28.332 had.28.333 had.28.334 had.28.335 had.28.336 had.28.337 had.28.338 had.28.339 had.28.340 had.28.341 had.28.342 had.28.343 had.28.344 had.28.345 had.28.346 had.28.347 had.28.348 had.28.349 had.28.350 had.28.351 had.28.352 had.28.353 had.28.354 had.28.355 had.28.356 had.28.357 had.28.358 had.28.359 had.28.360 had.28.361 had.28.362 had.28.363 had.28.364 had.28.365 had.28.366 had.28.367 had.28.368 had.28.369 had.28.370 had.28.371 had.28.372 had.28.373 had.28.374 had.28.375 had.28.376 had.28.377 had.28.378 had.28.379 had.28.380 had.28.381 had.28.382 had.28.383 had.28.384 had.28.385 had.28.386 had.28.387 had.28.388 had.28.389 had.28.390 had.28.391 had.28.392 had.28.393 had.28.394 had.28.395 had.28.396 had.28.397 had.28.398 had.28.399 had.28.400 had.28.401 had.28.402 had.28.403 had.28.404 had.28.405 had.28.406 had.28.407 had.28.408 had.28.409 had.28.410 had.28.411 had.28.412 had.28.413 had.28.414 had.28.415 had.28.416 had.28.417 had.28.418 had.28.419 had.28.420 had.28.421 had.28.422 had.28.423 had.28.424 had.28.425 had.28.426 had.28.427 had.28.428 had.28.429 had.28.430 had.28.431 had.28.432 had.28.433 had.28.434 had.28.435 had.28.436 had.28.437 had.28.438 had.28.439 had.28.440 had.28.441 had.28.442 had.28.443 had.28.444 had.28.445 had.28.446 had.28.447 had.28.448 had.28.449 had.28.450 had.28.451 had.28.452 had.28.453 had.28.454 had.28.455 had.28.456 had.28.457 had.28.458 had.28.459 had.28.460 had.28.461 had.28.462 had.28.463 had.28.464 had.28.465 had.28.466 had.28.467 had.28.468 had.28.469 had.28.470 had.28.471 had.28.472 had.28.473 had.28.474 had.28.475 had.28.476 had.28.477 had.28.478 had.28.479 had.28.480 had.28.481 had.28.482 had.28.483 had.28.484 had.28.485 had.28.486 had.28.487

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