MinT - Best Known (147−49, 147, s)-Nets in Base 8

Best Known (147−49, 147, s)-Nets in Base 8

(147−49, 147, 402)-Net over F₈ — Constructive and digital

Digital (98, 147, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 35, 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 (63, 112, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 56, 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]

(147−49, 147, 576)-Net in Base 8 — Constructive

(98, 147, 576)-net in base 8, using

t-expansion [i] based on (97, 147, 576)-net in base 8, using
- 7 times m-reduction [i] based on (97, 154, 576)-net in base 8, using
  - trace code for nets [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]

(147−49, 147, 1585)-Net over F₈ — Digital

Digital (98, 147, 1585)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(147−49, 147, 436563)-Net in Base 8 — Upper bound on s

There is no (98, 147, 436564)-net in base 8, because

1 times m-reduction [i] would yield (98, 146, 436564)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 709829 073521 685753 985133 441741 639693 597752 130781 556630 123853 688345 764940 984706 967390 304433 667695 660112 708350 868761 344965 136577 626472 > 8¹⁴⁶ [i]