R/hadamard-matrix-helpers.R

In bschneidr/svrep: Tools for Creating, Updating, and Analyzing Survey Replicate Weights

#' @title Find the smallest Hadamard order available in the 'survey' package
#' @description Identifies the order of the smallest Hadamard
#' matrix available in the 'survey' package, such that 
#' the order is greater than or equal to `n`.
#' This is useful for identifying the minimum number of
#' replicates that can be constructed for replication methods
#' such as BRR or Fay's generalized replication method.
#' @param n A single positive integer.
#' @return A single positive integer.
#' @details To get the Hadamard matrix of this size with the 'survey' package,
#' use `survey::hadamard(k - 1)`, where `k` is the
#' output of `find_minimum_hadamard_order()`.
#' @keywords internal
#' @md
find_minimum_hadamard_order <- function(n) {
  
  # Sizes of Hadamard matrices included in the 'survey' package
  included_hadamard_sizes <- c(2, 4, 16, 28, 36)
  
  # Prime numbers p such that p + 1 is a multiple of 4
  small_primes <- c(
    3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 
    79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 
    191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 
    311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 
    463, 467, 479, 487, 491, 499, 503, 523, 547, 563, 571, 
    587, 599, 607, 619, 631, 643, 647, 659, 683, 691, 719, 
    727, 739, 743, 751, 787, 811, 823, 827, 839, 859, 863, 
    883, 887, 907, 911, 919, 947, 967, 971, 983, 991, 1019, 
    1031, 1039, 1051, 1063, 1087, 1091, 1103, 1123, 1151, 
    1163, 1171, 1187, 1223, 1231, 1259, 1279, 1283, 1291, 
    1303, 1307, 1319, 1327, 1367, 1399, 1423, 1427, 1439, 
    1447, 1451, 1459, 1471, 1483, 1487, 1499, 1511, 1523, 
    1531, 1543, 1559, 1567, 1571, 1579, 1583, 1607, 1619, 
    1627, 1663, 1667, 1699, 1723, 1747, 1759, 1783, 1787, 
    1811, 1823, 1831, 1847, 1867, 1871, 1879, 1907, 1931, 
    1951, 1979, 1987, 1999, 2003, 2011, 2027, 2039, 2063, 
    2083, 2087, 2099, 2111, 2131, 2143, 2179, 2203, 2207, 
    2239, 2243, 2251, 2267, 2287, 2311, 2339, 2347, 2351, 
    2371, 2383, 2399, 2411, 2423, 2447, 2459, 2467, 2503, 
    2531, 2539, 2543, 2551, 2579, 2591, 2647, 2659, 2663, 
    2671, 2683, 2687, 2699, 2707, 2711, 2719, 2731, 2767, 
    2791, 2803, 2819, 2843, 2851, 2879, 2887, 2903, 2927, 
    2939, 2963, 2971, 2999, 3011, 3019, 3023, 3067, 3079, 
    3083, 3119, 3163, 3167, 3187, 3191, 3203, 3251, 3259, 
    3271, 3299, 3307, 3319, 3323, 3331, 3343, 3347, 3359, 
    3371, 3391, 3407, 3463, 3467, 3491, 3499, 3511, 3527, 
    3539, 3547, 3559, 3571, 3583, 3607, 3623, 3631, 3643, 
    3659, 3671, 3691, 3719, 3727, 3739, 3767, 3779, 3803, 
    3823, 3847, 3851, 3863, 3907, 3911, 3919, 3923, 3931, 
    3943, 3947, 3967, 4003, 4007, 4019, 4027, 4051, 4079, 
    4091, 4099, 4111, 4127, 4139, 4159, 4211, 4219, 4231, 
    4243, 4259, 4271, 4283, 4327, 4339, 4363, 4391, 4423, 
    4447, 4451, 4463, 4483, 4507, 4519, 4523, 4547, 4567, 
    4583, 4591, 4603, 4639, 4643, 4651, 4663, 4679, 4691, 
    4703, 4723, 4751, 4759, 4783, 4787, 4799, 4831, 4871, 
    4903, 4919, 4931, 4943, 4951, 4967, 4987, 4999, 5003, 
    5011, 5023, 5039, 5051, 5059, 5087, 5099, 5107, 5119, 
    5147, 5167, 5171, 5179, 5227, 5231, 5279, 5303, 5323, 
    5347, 5351, 5387, 5399, 5407, 5419, 5431, 5443, 5471, 
    5479, 5483, 5503, 5507, 5519, 5527, 5531, 5563, 5591, 
    5623, 5639, 5647, 5651, 5659, 5683, 5711, 5743, 5779, 
    5783, 5791, 5807, 5827, 5839, 5843, 5851, 5867, 5879, 
    5903, 5923, 5927, 5939, 5987, 6007, 6011, 6043, 6047, 
    6067, 6079, 6091, 6131, 6143, 6151, 6163, 6199, 6203, 
    6211, 6247, 6263, 6271, 6287, 6299, 6311, 6323, 6343, 
    6359, 6367, 6379, 6427, 6451, 6491, 6547, 6551, 6563, 
    6571, 6599, 6607, 6619, 6659, 6679, 6691, 6703, 6719, 
    6763, 6779, 6791, 6803, 6823, 6827, 6863, 6871, 6883, 
    6899, 6907, 6911, 6947, 6959, 6967, 6971, 6983, 6991, 
    7019, 7027, 7039, 7043, 7079, 7103, 7127, 7151, 7159, 
    7187, 7207, 7211, 7219, 7243, 7247, 7283, 7307, 7331, 
    7351, 7411, 7451, 7459, 7487, 7499, 7507, 7523, 7547, 
    7559, 7583, 7591, 7603, 7607, 7639, 7643, 7687, 7691, 
    7699, 7703, 7723, 7727, 7759, 7823, 7867, 7879, 7883, 
    7907, 7919
  )
  
  # The 'survey' package can use an included Hadamard matrix
  # or construct one using the Paley construction
  starting_hadamard_sizes <- c(included_hadamard_sizes, small_primes + 1)
  
  # From a starting Hadamard matrix of dimension p, another Hadamard matrix
  # can be created of dimension (p + 1)*2^k, for nonnegative integer k
  candidate_sizes <- sapply(starting_hadamard_sizes, FUN = \(start_size) {
    start_size * 2^(pmax(0, ceiling(log2(n) - log2(start_size))))
  })
  
  min_hadamard_size <- min(candidate_sizes)
  return(min_hadamard_size)
}