MinT - Best Known (28, 58, s)-Nets in Base 64

Best Known (28, 58, s)-Nets in Base 64

(28, 58, 513)-Net over F₆₄ — Constructive and digital

Digital (28, 58, 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]

(28, 58, 840)-Net over F₆₄ — Digital

Digital (28, 58, 840)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁸, 840, F₆₄, 30) (dual of [840, 782, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁵⁸, 1365, F₆₄, 30) (dual of [1365, 1307, 31]-code), using
  - the BCH-code C(I) with length 1365Â | 64²âˆ’1, defining interval IÂ =Â {1,24,47,…,668}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(28, 58, 982468)-Net in Base 64 — Upper bound on s

There is no (28, 58, 982469)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 573 375936 250545 675876 661006 314706 405519 726018 567242 556188 621346 244717 813319 221217 608041 128318 169202 817472 > 64⁵⁸ [i]