MinT - Best Known (66−37, 66, s)-Nets in Base 128

Best Known (66−37, 66, s)-Nets in Base 128

(66−37, 66, 438)-Net over F₁₂₈ — Constructive and digital

Digital (29, 66, 438)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (29, 67, 438)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 20, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. digital (9, 47, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(66−37, 66, 516)-Net in Base 128 — Constructive

(29, 66, 516)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 22, 258)-net in base 128, using
  - 2 times m-reduction [i] based on (4, 24, 258)-net in base 128, using
    - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. (7, 44, 258)-net in base 128, using
  - 4 times m-reduction [i] based on (7, 48, 258)-net in base 128, using
    - base change [i] based on digital (1, 42, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(66−37, 66, 839)-Net over F₁₂₈ — Digital

Digital (29, 66, 839)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(66−37, 66, 2419324)-Net in Base 128 — Upper bound on s

There is no (29, 66, 2419325)-net in base 128, because

1 times m-reduction [i] would yield (29, 65, 2419325)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 93035 718941 523841 926460 978362 277715 269817 322483 727236 415982 814780 777692 484353 352203 207244 879049 063698 049688 033115 779348 050389 622577 142146 > 128⁶⁵ [i]