MinT - Best Known (160−54, 160, s)-Nets in Base 8

Best Known (160−54, 160, s)-Nets in Base 8

(160−54, 160, 402)-Net over F₈ — Constructive and digital

Digital (106, 160, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 38, 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 (68, 122, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 61, 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]

(160−54, 160, 576)-Net in Base 8 — Constructive

(106, 160, 576)-net in base 8, using

t-expansion [i] based on (105, 160, 576)-net in base 8, using
- 8 times m-reduction [i] based on (105, 168, 576)-net in base 8, using
  - trace code for nets [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]

(160−54, 160, 1593)-Net over F₈ — Digital

Digital (106, 160, 1593)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(160−54, 160, 350697)-Net in Base 8 — Upper bound on s

There is no (106, 160, 350698)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 121955 515027 709393 273819 977797 244280 881965 060882 583333 202125 372460 428748 248166 910500 714386 254079 434212 714501 709560 546451 909325 589506 214691 186384 > 8¹⁶⁰ [i]