MinT - Best Known (28, 28+31, s)-Nets in Base 128

Best Known (28, 28+31, s)-Nets in Base 128

(28, 28+31, 480)-Net over F₁₂₈ — Constructive and digital

Digital (28, 59, 480)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (28, 60, 480)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 19, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, 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]

(28, 28+31, 546)-Net in Base 128 — Constructive

(28, 59, 546)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 19, 258)-net in base 128, using
  - 5 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. digital (9, 40, 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]

(28, 28+31, 1337)-Net over F₁₂₈ — Digital

Digital (28, 59, 1337)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(28, 28+31, 7109532)-Net in Base 128 — Upper bound on s

There is no (28, 59, 7109533)-net in base 128, because

1 times m-reduction [i] would yield (28, 58, 7109533)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 165 264221 262310 764945 695257 510084 549125 612257 039406 310010 231310 061247 808304 952284 956573 612333 629695 105733 575663 887293 355536 > 128⁵⁸ [i]