MinT - Best Known (34, 34+38, s)-Nets in Base 128

Best Known (34, 34+38, s)-Nets in Base 128

(34, 34+38, 504)-Net over F₁₂₈ — Constructive and digital

Digital (34, 72, 504)-net over F₁₂₈, using

2 times m-reduction [i] based on digital (34, 74, 504)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 25, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]
  2. digital (9, 49, 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]

(34, 34+38, 547)-Net in Base 128 — Constructive

(34, 72, 547)-net in base 128, using

t-expansion [i] based on (33, 72, 547)-net in base 128, using
- (u,Â u+v)-construction [i] based on
  1. (5, 24, 259)-net in base 128, using
    - base change [i] based on digital (2, 21, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
  2. digital (9, 48, 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]

(34, 34+38, 1473)-Net over F₁₂₈ — Digital

Digital (34, 72, 1473)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 34+38, 6034313)-Net in Base 128 — Upper bound on s

There is no (34, 72, 6034314)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 52 374367 169921 838170 848195 968997 351455 979700 911737 817900 060639 506485 291514 799620 117304 858114 077578 978294 464812 160374 438283 106672 975934 271464 936503 404768 > 128⁷² [i]