MinT - Best Known (63−19, 63, s)-Nets in Base 8

Best Known (63−19, 63, s)-Nets in Base 8

(63−19, 63, 371)-Net over F₈ — Constructive and digital

Digital (44, 63, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (33, 52, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 26, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(63−19, 63, 531)-Net in Base 8 — Constructive

(44, 63, 531)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. (33, 52, 514)-net in base 8, using
  - base change [i] based on digital (20, 39, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (20, 40, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 20, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(63−19, 63, 1572)-Net over F₈ — Digital

Digital (44, 63, 1572)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶³, 1572, F₈, 19) (dual of [1572, 1509, 20]-code), using
- 1508 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 2, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 7 times 0, 1, 7 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 13 times 0, 1, 15 times 0, 1, 17 times 0, 1, 19 times 0, 1, 22 times 0, 1, 25 times 0, 1, 28 times 0, 1, 32 times 0, 1, 36 times 0, 1, 41 times 0, 1, 46 times 0, 1, 51 times 0, 1, 58 times 0, 1, 66 times 0, 1, 74 times 0, 1, 83 times 0, 1, 94 times 0, 1, 105 times 0, 1, 119 times 0, 1, 133 times 0, 1, 150 times 0, 1, 169 times 0) [i] based on linear OA(8¹⁹, 20, F₈, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,8)), using
  - dual of repetition code with length 20 [i]

(63−19, 63, 986137)-Net in Base 8 — Upper bound on s

There is no (44, 63, 986138)-net in base 8, because

1 times m-reduction [i] would yield (44, 62, 986138)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 98 079789 763732 370080 367707 965765 704931 320771 052476 019153 > 8⁶² [i]