MinT - Best Known (59−23, 59, s)-Nets in Base 27

Best Known (59−23, 59, s)-Nets in Base 27

(59−23, 59, 240)-Net over F₂₇ — Constructive and digital

Digital (36, 59, 240)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 7, 56)-net over F₂₇, using
  - s-reduction based on digital (2, 7, 351)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁷, 351, F₂₇, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (3, 10, 56)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 7, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
3. digital (4, 15, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
4. digital (4, 27, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(59−23, 59, 370)-Net in Base 27 — Constructive

(36, 59, 370)-net in base 27, using

21 times m-reduction [i] based on (36, 80, 370)-net in base 27, using
- base change [i] based on digital (16, 60, 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]

(59−23, 59, 2413)-Net over F₂₇ — Digital

Digital (36, 59, 2413)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−23, 59, 6656274)-Net in Base 27 — Upper bound on s

There is no (36, 59, 6656275)-net in base 27, because

1 times m-reduction [i] would yield (36, 58, 6656275)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 104495 796527 216471 561181 514957 656868 019457 079723 213940 978859 448824 395731 241443 118451 > 27⁵⁸ [i]

Best Known (59−23, 59, s)-Nets in Base 27

(59−23, 59, 240)-Net over F27 — Constructive and digital

(59−23, 59, 370)-Net in Base 27 — Constructive

(59−23, 59, 2413)-Net over F27 — Digital

(59−23, 59, 6656274)-Net in Base 27 — Upper bound on s

(59−23, 59, 240)-Net over F₂₇ — Constructive and digital

(59−23, 59, 2413)-Net over F₂₇ — Digital