MinT - Best Known (72, 173, s)-Nets in Base 8

Best Known (72, 173, s)-Nets in Base 8

(72, 173, 100)-Net over F₈ — Constructive and digital

Digital (72, 173, 100)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 58, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [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 (14, 115, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(72, 173, 149)-Net over F₈ — Digital

Digital (72, 173, 149)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 173, 156)-Net in Base 8

(72, 173, 156)-net in base 8, using

8¹ times duplication [i] based on (71, 172, 156)-net in base 8, using
- t-expansion [i] based on (70, 172, 156)-net in base 8, using
  - base change [i] based on digital (27, 129, 156)-net over F₁₆, using
    - net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
        a curve from the manYPoints database [i]

(72, 173, 3526)-Net in Base 8 — Upper bound on s

There is no (72, 173, 3527)-net in base 8, because

1 times m-reduction [i] would yield (72, 172, 3527)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 216534 864114 319352 234284 957669 159384 383472 885511 938256 732113 540957 994607 485843 311212 835232 441495 742744 608591 684559 818498 900433 705669 457768 483398 379168 051836 > 8¹⁷² [i]