MinT - Best Known (107−42, 107, s)-Nets in Base 27

Best Known (107−42, 107, s)-Nets in Base 27

(107−42, 107, 282)-Net over F₂₇ — Constructive and digital

Digital (65, 107, 282)-net over F₂₇, using

1 times m-reduction [i] based on digital (65, 108, 282)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (10, 24, 94)-net over F₂₇, using
    - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
        a function field by SÃ©mirat [i]
  2. digital (10, 31, 94)-net over F₂₇, using
    - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)
  3. digital (10, 53, 94)-net over F₂₇, using
    - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(107−42, 107, 730)-Net in Base 27 — Constructive

(65, 107, 730)-net in base 27, using

t-expansion [i] based on (63, 107, 730)-net in base 27, using
- 1 times m-reduction [i] based on (63, 108, 730)-net in base 27, using
  - base change [i] based on digital (36, 81, 730)-net over F₈₁, using
    - net from sequence [i] based on digital (36, 729)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 36 and N(F) ≥ 730, using
        the Hermitian function field over F₈₁ [i]

(107−42, 107, 3397)-Net over F₂₇ — Digital

Digital (65, 107, 3397)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(107−42, 107, 6556282)-Net in Base 27 — Upper bound on s

There is no (65, 107, 6556283)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1431 933835 651213 766763 355159 685356 309356 148855 639924 381880 173584 626538 706481 429448 784974 483423 706731 197795 955230 055601 351600 322608 800923 384581 425178 216887 > 27¹⁰⁷ [i]