MinT - Best Known (26, 59, s)-Nets in Base 128

Best Known (26, 59, s)-Nets in Base 128

(26, 59, 438)-Net over F₁₂₈ — Constructive and digital

Digital (26, 59, 438)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 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, 42, 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]

(26, 59, 516)-Net in Base 128 — Constructive

(26, 59, 516)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 20, 258)-net in base 128, using
  - 4 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. (6, 39, 258)-net in base 128, using
  - 1 times m-reduction [i] based on (6, 40, 258)-net in base 128, using
    - base change [i] based on digital (1, 35, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(26, 59, 789)-Net over F₁₂₈ — Digital

Digital (26, 59, 789)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(26, 59, 2330074)-Net in Base 128 — Upper bound on s

There is no (26, 59, 2330075)-net in base 128, because

1 times m-reduction [i] would yield (26, 58, 2330075)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 165 265075 874791 057401 956798 409523 967109 402295 923295 463823 693958 834000 650472 173521 637926 867881 617192 699768 186295 963338 443571 > 128⁵⁸ [i]