MinT - Best Known (30−16, 30, s)-Nets in Base 64

Best Known (30−16, 30, s)-Nets in Base 64

(30−16, 30, 210)-Net over F₆₄ — Constructive and digital

Digital (14, 30, 210)-net over F₆₄, using

1 times m-reduction [i] based on digital (14, 31, 210)-net over F₆₄, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 5, 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. digital (0, 8, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
  3. digital (1, 18, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]

(30−16, 30, 322)-Net in Base 64 — Constructive

(14, 30, 322)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 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, 22, 257)-net in base 64, using
  - 2 times m-reduction [i] based on (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]

(30−16, 30, 523)-Net over F₆₄ — Digital

Digital (14, 30, 523)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³⁰, 523, F₆₄, 16) (dual of [523, 493, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(64³⁰, 819, F₆₄, 16) (dual of [819, 789, 17]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 819Â | 64²âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(30−16, 30, 354421)-Net in Base 64 — Upper bound on s

There is no (14, 30, 354422)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 532515 241862 232429 908762 060695 880508 205932 528493 785918 > 64³⁰ [i]