MinT - Best Known (58, 58+45, s)-Nets in Base 27

Best Known (58, 58+45, s)-Nets in Base 27

(58, 58+45, 246)-Net over F₂₇ — Constructive and digital

Digital (58, 103, 246)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 22, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (7, 29, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (7, 52, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(58, 58+45, 370)-Net in Base 27 — Constructive

(58, 103, 370)-net in base 27, using

t-expansion [i] based on (43, 103, 370)-net in base 27, using
- 5 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(58, 58+45, 1510)-Net over F₂₇ — Digital

Digital (58, 103, 1510)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 58+45, 1507298)-Net in Base 27 — Upper bound on s

There is no (58, 103, 1507299)-net in base 27, because

1 times m-reduction [i] would yield (58, 102, 1507299)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 99 794366 611213 482098 474073 297962 814577 061521 377426 252376 838744 715046 216233 673221 365444 680653 772623 627247 253199 424322 931025 334762 471789 598316 454189 > 27¹⁰² [i]