MinT - Best Known (4, 4+7, s)-Nets in Base 64

Best Known (4, 4+7, s)-Nets in Base 64

(4, 4+7, 145)-Net over F₆₄ — Constructive and digital

Digital (4, 11, 145)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 3, 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 (1, 8, 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]

(4, 4+7, 168)-Net over F₆₄ — Digital

Digital (4, 11, 168)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64¹¹, 168, F₆₄, 7) (dual of [168, 157, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(64¹¹, 195, F₆₄, 7) (dual of [195, 184, 8]-code), using
  - the BCH-code C(I) with length 195Â | 64²âˆ’1, defining interval IÂ =Â {64,86,108,…,1}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(4, 4+7, 258)-Net in Base 64 — Constructive

(4, 11, 258)-net in base 64, using

1 times m-reduction [i] based on (4, 12, 258)-net in base 64, using
- base change [i] based on digital (1, 9, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(4, 4+7, 289)-Net in Base 64

(4, 11, 289)-net in base 64, using

1 times m-reduction [i] based on (4, 12, 289)-net in base 64, using
- base change [i] based on digital (1, 9, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(4, 4+7, 30243)-Net in Base 64 — Upper bound on s

There is no (4, 11, 30244)-net in base 64, because

1 times m-reduction [i] would yield (4, 10, 30244)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 153010 375277 047407 > 64¹⁰ [i]