MinT - Best Known (148−53, 148, s)-Nets in Base 8

Best Known (148−53, 148, s)-Nets in Base 8

(148−53, 148, 371)-Net over F₈ — Constructive and digital

Digital (95, 148, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 28, 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 (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]

(148−53, 148, 576)-Net in Base 8 — Constructive

(95, 148, 576)-net in base 8, using

2 times m-reduction [i] based on (95, 150, 576)-net in base 8, using
- trace code for nets [i] based on (20, 75, 288)-net in base 64, using
  - 2 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]

(148−53, 148, 1100)-Net over F₈ — Digital

Digital (95, 148, 1100)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(148−53, 148, 192349)-Net in Base 8 — Upper bound on s

There is no (95, 148, 192350)-net in base 8, because

1 times m-reduction [i] would yield (95, 147, 192350)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5 679089 904193 582244 723843 221755 234739 986534 803310 377299 757843 047186 034149 652015 953421 119280 834617 394673 546040 433265 699274 984946 687534 > 8¹⁴⁷ [i]