MinT - Best Known (69−39, 69, s)-Nets in Base 128

Best Known (69−39, 69, s)-Nets in Base 128

(69−39, 69, 438)-Net over F₁₂₈ — Constructive and digital

Digital (30, 69, 438)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (30, 70, 438)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 21, 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, 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]

(69−39, 69, 516)-Net in Base 128 — Constructive

(30, 69, 516)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 23, 258)-net in base 128, using
  - 1 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. (7, 46, 258)-net in base 128, using
  - 2 times m-reduction [i] based on (7, 48, 258)-net in base 128, using
    - base change [i] based on digital (1, 42, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(69−39, 69, 812)-Net over F₁₂₈ — Digital

Digital (30, 69, 812)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−39, 69, 2172718)-Net in Base 128 — Upper bound on s

There is no (30, 69, 2172719)-net in base 128, because

1 times m-reduction [i] would yield (30, 68, 2172719)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 195109 540960 054447 288454 661762 668313 698074 444025 898980 896140 406164 820300 605880 008082 106687 115199 625877 312793 405278 584070 594316 805668 841866 372200 > 128⁶⁸ [i]