MinT - Best Known (62−32, 62, s)-Nets in Base 128

Best Known (62−32, 62, s)-Nets in Base 128

(62−32, 62, 504)-Net over F₁₂₈ — Constructive and digital

Digital (30, 62, 504)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 21, 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, 41, 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]

(62−32, 62, 547)-Net in Base 128 — Constructive

(30, 62, 547)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (5, 21, 259)-net in base 128, using
  - 3 times m-reduction [i] based on (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, 41, 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]

(62−32, 62, 1617)-Net over F₁₂₈ — Digital

Digital (30, 62, 1617)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(62−32, 62, 7837421)-Net in Base 128 — Upper bound on s

There is no (30, 62, 7837422)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 44362 744833 987094 194401 950625 091798 743619 189975 949314 365554 034218 994821 646167 595167 550509 869946 852633 895235 740168 757903 459541 371590 > 128⁶² [i]