MinT - Best Known (63, 63+41, s)-Nets in Base 27

Best Known (63, 63+41, s)-Nets in Base 27

(63, 63+41, 282)-Net over F₂₇ — Constructive and digital

Digital (63, 104, 282)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (10, 23, 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, 30, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)
3. digital (10, 51, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(63, 63+41, 730)-Net in Base 27 — Constructive

(63, 104, 730)-net in base 27, using

4 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]

(63, 63+41, 3214)-Net over F₂₇ — Digital

Digital (63, 104, 3214)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(63, 63+41, 7513731)-Net in Base 27 — Upper bound on s

There is no (63, 104, 7513732)-net in base 27, because

1 times m-reduction [i] would yield (63, 103, 7513732)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2694 438189 396663 505962 483899 235674 384282 745648 882947 231126 312729 969450 337829 636614 001208 833879 293860 255266 903029 305958 205645 784292 836583 968910 024513 > 27¹⁰³ [i]