MinT - Best Known (72, 72+30, s)-Nets in Base 8

Best Known (72, 72+30, s)-Nets in Base 8

(72, 72+30, 416)-Net over F₈ — Constructive and digital

Digital (72, 102, 416)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (21, 36, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 18, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
2. digital (36, 66, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 33, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄ (see above)

(72, 72+30, 576)-Net in Base 8 — Constructive

(72, 102, 576)-net in base 8, using

8 times m-reduction [i] based on (72, 110, 576)-net in base 8, using
- trace code for nets [i] based on (17, 55, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 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]

(72, 72+30, 2517)-Net over F₈ — Digital

Digital (72, 102, 2517)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+30, 1269630)-Net in Base 8 — Upper bound on s

There is no (72, 102, 1269631)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 130 370901 333530 254040 840016 359337 622069 037669 093121 552212 518937 200221 887261 534411 661365 910936 > 8¹⁰² [i]