MinT - Best Known (15, 15+18, s)-Nets in Base 64

Best Known (15, 15+18, s)-Nets in Base 64

(15, 15+18, 208)-Net over F₆₄ — Constructive and digital

Digital (15, 33, 208)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 12, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]
2. digital (3, 21, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄ (see above)

(15, 15+18, 322)-Net in Base 64 — Constructive

(15, 33, 322)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
2. (6, 24, 257)-net in base 64, using
  - base change [i] based on digital (0, 18, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(15, 15+18, 435)-Net over F₆₄ — Digital

Digital (15, 33, 435)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³³, 435, F₆₄, 18) (dual of [435, 402, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(64³³, 455, F₆₄, 18) (dual of [455, 422, 19]-code), using
  - the BCH-code C(I) with length 455Â | 64²âˆ’1, defining interval IÂ =Â {1,9,17,…,137}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(15, 15+18, 276098)-Net in Base 64 — Upper bound on s

There is no (15, 33, 276099)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 401739 064408 020279 590060 484508 382947 376730 377615 382451 064224 > 64³³ [i]