MinT - Best Known (58, 58+36, s)-Nets in Base 27

Best Known (58, 58+36, s)-Nets in Base 27

(58, 58+36, 274)-Net over F₂₇ — Constructive and digital

Digital (58, 94, 274)-net over F₂₇, using

1 times m-reduction [i] based on digital (58, 95, 274)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 13, 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]
  2. digital (4, 16, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  3. digital (4, 22, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  4. digital (7, 44, 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]

(58, 58+36, 452)-Net in Base 27 — Constructive

(58, 94, 452)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (6, 24, 82)-net in base 27, using
  - base change [i] based on digital (0, 18, 82)-net over F₈₁, using
    - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
      - Niederreiter sequence [i]
2. (34, 70, 370)-net in base 27, using
  - 2 times m-reduction [i] based on (34, 72, 370)-net in base 27, using
    - base change [i] based on digital (16, 54, 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+36, 3755)-Net over F₂₇ — Digital

Digital (58, 94, 3755)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁹⁴, 3755, F₂₇, 36) (dual of [3755, 3661, 37]-code), using
- 3660 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 16 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 25 times 0, 1, 27 times 0, 1, 30 times 0, 1, 33 times 0, 1, 36 times 0, 1, 40 times 0, 1, 45 times 0, 1, 49 times 0, 1, 54 times 0, 1, 60 times 0, 1, 65 times 0, 1, 73 times 0, 1, 80 times 0, 1, 88 times 0, 1, 96 times 0, 1, 107 times 0, 1, 117 times 0, 1, 129 times 0, 1, 142 times 0, 1, 156 times 0, 1, 172 times 0, 1, 188 times 0, 1, 208 times 0, 1, 229 times 0, 1, 251 times 0, 1, 276 times 0, 1, 304 times 0, 1, 334 times 0) [i] based on linear OA(27³⁶, 37, F₂₇, 36) (dual of [37, 1, 37]-code or 37-arc in PG(35,27)), using
  - dual of repetition code with length 37 [i]

(58, 58+36, large)-Net in Base 27 — Upper bound on s

There is no (58, 94, large)-net in base 27, because

34 times m-reduction [i] would yield (58, 60, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]