MinT - Best Known (157−53, 157, s)-Nets in Base 8

Best Known (157−53, 157, s)-Nets in Base 8

(157−53, 157, 402)-Net over F₈ — Constructive and digital

Digital (104, 157, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 37, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (67, 120, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 60, 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]

(157−53, 157, 576)-Net in Base 8 — Constructive

(104, 157, 576)-net in base 8, using

9 times m-reduction [i] based on (104, 166, 576)-net in base 8, using
- trace code for nets [i] based on (21, 83, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(157−53, 157, 1566)-Net over F₈ — Digital

Digital (104, 157, 1566)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−53, 157, 395109)-Net in Base 8 — Upper bound on s

There is no (104, 157, 395110)-net in base 8, because

1 times m-reduction [i] would yield (104, 156, 395110)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 762 187705 068420 236419 897573 493106 359353 078552 298397 822246 801948 545300 705155 650232 473020 506381 645943 730323 660223 370586 564650 132197 468670 690216 > 8¹⁵⁶ [i]