MinT - Best Known (16, 16+20, s)-Nets in Base 128

Best Known (16, 16+20, s)-Nets in Base 128

(16, 16+20, 387)-Net over F₁₂₈ — Constructive and digital

Digital (16, 36, 387)-net over F₁₂₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 6, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (0, 10, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)
3. digital (0, 20, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)

(16, 16+20, 515)-Net in Base 128 — Constructive

(16, 36, 515)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (2, 12, 257)-net in base 128, using
  - 4 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
    - base change [i] based on digital (0, 14, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. (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]

(16, 16+20, 623)-Net over F₁₂₈ — Digital

Digital (16, 36, 623)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(16, 16+20, 1374443)-Net in Base 128 — Upper bound on s

There is no (16, 36, 1374444)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 7237 009488 343931 483186 228108 218491 851585 340576 184839 203309 316830 092711 966967 > 128³⁶ [i]