MinT - Best Known (18, 18+21, s)-Nets in Base 64

Best Known (18, 18+21, s)-Nets in Base 64

Digital (18, 39, 257)-net over F₆₄, using

(18, 39, 337)-net in base 64, using

Digital (18, 39, 507)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³⁹, 507, F₆₄, 21) (dual of [507, 468, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(64³⁹, 585, F₆₄, 21) (dual of [585, 546, 22]-code), using
  - the BCH-code C(I) with length 585Â | 64²âˆ’1, defining interval IÂ =Â {1,60,119,…,11}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(18, 39, 513)-net in base 64, using

1 times m-reduction [i] based on (18, 40, 513)-net in base 64, using
- base change [i] based on digital (8, 30, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

There is no (18, 39, 524948)-net in base 64, because

1 times m-reduction [i] would yield (18, 38, 524948)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 431 363960 497513 626377 714493 383984 663975 314468 200011 275427 577366 087466 > 64³⁸ [i]