MinT - Best Known (31, 72, s)-Nets in Base 128

Best Known (31, 72, s)-Nets in Base 128

(31, 72, 438)-Net over F₁₂₈ — Constructive and digital

Digital (31, 72, 438)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (31, 73, 438)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 22, 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, 51, 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]

(31, 72, 516)-Net in Base 128 — Constructive

(31, 72, 516)-net in base 128, using

base change [i] based on digital (22, 63, 516)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. 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 (1, 42, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(31, 72, 791)-Net over F₁₂₈ — Digital

Digital (31, 72, 791)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 72, 1977405)-Net in Base 128 — Upper bound on s

There is no (31, 72, 1977406)-net in base 128, because

1 times m-reduction [i] would yield (31, 71, 1977406)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 409175 079136 625372 075750 171325 395289 169607 891332 610329 165138 349335 163858 433148 591843 303115 352172 829511 290862 517147 465369 030763 509088 962927 129907 408227 > 128⁷¹ [i]