MinT - Best Known (38, 38+40, s)-Nets in Base 64

Best Known (38, 38+40, s)-Nets in Base 64

(38, 38+40, 513)-Net over F₆₄ — Constructive and digital

Digital (38, 78, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 78, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(38, 38+40, 1072)-Net over F₆₄ — Digital

Digital (38, 78, 1072)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁷⁸, 1072, F₆₄, 40) (dual of [1072, 994, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁷⁸, 1365, F₆₄, 40) (dual of [1365, 1287, 41]-code), using
  - the BCH-code C(I) with length 1365Â | 64²âˆ’1, defining interval IÂ =Â {1,12,23,…,430}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]

(38, 38+40, 1459031)-Net in Base 64 — Upper bound on s

There is no (38, 78, 1459032)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 762 151237 304477 197323 609675 250610 675979 752343 289658 041657 514109 080223 626772 375925 923520 249128 571590 871691 727136 919692 460611 485727 823147 404879 > 64⁷⁸ [i]