MinT - Best Known (63, 63+42, s)-Nets in Base 27

Best Known (63, 63+42, s)-Nets in Base 27

(63, 63+42, 272)-Net over F₂₇ — Constructive and digital

Digital (63, 105, 272)-net over F₂₇, using

1 times m-reduction [i] based on digital (63, 106, 272)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (8, 22, 84)-net over F₂₇, using
    - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, 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, 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]
  3. digital (10, 53, 94)-net over F₂₇, using
    - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(63, 63+42, 730)-Net in Base 27 — Constructive

(63, 105, 730)-net in base 27, using

3 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+42, 2895)-Net over F₂₇ — Digital

Digital (63, 105, 2895)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(63, 63+42, 4790017)-Net in Base 27 — Upper bound on s

There is no (63, 105, 4790018)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 964251 200727 750317 945174 625243 067866 224589 812771 563456 884069 680733 948523 563672 698458 897568 585423 695328 703745 878087 805753 925117 148702 306552 735310 777133 > 27¹⁰⁵ [i]