MinT - Best Known (103, 103+55, s)-Nets in Base 8

Best Known (103, 103+55, s)-Nets in Base 8

(103, 103+55, 388)-Net over F₈ — Constructive and digital

Digital (103, 158, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 34, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (69, 124, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 62, 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]

(103, 103+55, 576)-Net in Base 8 — Constructive

(103, 158, 576)-net in base 8, using

6 times m-reduction [i] based on (103, 164, 576)-net in base 8, using
- trace code for nets [i] based on (21, 82, 288)-net in base 64, using
  - 2 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]

(103, 103+55, 1342)-Net over F₈ — Digital

Digital (103, 158, 1342)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(103, 103+55, 278345)-Net in Base 8 — Upper bound on s

There is no (103, 158, 278346)-net in base 8, because

1 times m-reduction [i] would yield (103, 157, 278346)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6097 741146 018734 868818 199039 037337 981055 493984 663063 530782 236072 659241 630000 999109 802908 492361 919612 704531 660586 357648 266793 058438 096251 646496 > 8¹⁵⁷ [i]