MinT - Best Known (77−41, 77, s)-Nets in Base 32

Best Known (77−41, 77, s)-Nets in Base 32

(77−41, 77, 202)-Net over F₃₂ — Constructive and digital

Digital (36, 77, 202)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 27, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]
2. digital (9, 50, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(77−41, 77, 288)-Net in Base 32 — Constructive

(36, 77, 288)-net in base 32, using

t-expansion [i] based on (35, 77, 288)-net in base 32, using
- 14 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
  - base change [i] based on digital (9, 65, 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]

(77−41, 77, 409)-Net over F₃₂ — Digital

Digital (36, 77, 409)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−41, 77, 140437)-Net in Base 32 — Upper bound on s

There is no (36, 77, 140438)-net in base 32, because

1 times m-reduction [i] would yield (36, 76, 140438)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 462801 505339 239911 386084 059636 758042 885869 628106 664245 173781 447639 459349 183630 326330 896018 777824 415275 665837 962669 > 32⁷⁶ [i]