MinT - Best Known (31−17, 31, s)-Nets in Base 64

Best Known (31−17, 31, s)-Nets in Base 64

(31−17, 31, 210)-Net over F₆₄ — Constructive and digital

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]

(31−17, 31, 322)-Net in Base 64 — Constructive

(14, 31, 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, 23, 257)-net in base 64, using
  - 1 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]

(31−17, 31, 411)-Net over F₆₄ — Digital

Digital (14, 31, 411)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³¹, 411, F₆₄, 17) (dual of [411, 380, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(64³¹, 455, F₆₄, 17) (dual of [455, 424, 18]-code), using
  - the BCH-code C(I) with length 455Â | 64²âˆ’1, defining interval IÂ =Â {1,9,17,…,129}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(31−17, 31, 354421)-Net in Base 64 — Upper bound on s

There is no (14, 31, 354422)-net in base 64, because

1 times m-reduction [i] would yield (14, 30, 354422)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 532515 241862 232429 908762 060695 880508 205932 528493 785918 > 64³⁰ [i]