MinT - Best Known (144−50, 144, s)-Nets in Base 8

Best Known (144−50, 144, s)-Nets in Base 8

(144−50, 144, 382)-Net over F₈ — Constructive and digital

Digital (94, 144, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 30, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (64, 114, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 57, 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]

(144−50, 144, 576)-Net in Base 8 — Constructive

(94, 144, 576)-net in base 8, using

4 times m-reduction [i] based on (94, 148, 576)-net in base 8, using
- trace code for nets [i] based on (20, 74, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(144−50, 144, 1255)-Net over F₈ — Digital

Digital (94, 144, 1255)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(144−50, 144, 231364)-Net in Base 8 — Upper bound on s

There is no (94, 144, 231365)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11091 174511 433307 310441 807777 717326 335671 670419 401005 728192 506278 310549 593481 664221 655566 066099 124679 196096 541234 109745 243778 738656 > 8¹⁴⁴ [i]