MinT - Best Known (126−67, 126, s)-Nets in Base 8

Best Known (126−67, 126, s)-Nets in Base 8

(126−67, 126, 113)-Net over F₈ — Constructive and digital

Digital (59, 126, 113)-net over F₈, using

1 times m-reduction [i] based on digital (59, 127, 113)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 45, 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 (14, 82, 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]

(126−67, 126, 164)-Net over F₈ — Digital

Digital (59, 126, 164)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−67, 126, 4934)-Net in Base 8 — Upper bound on s

There is no (59, 126, 4935)-net in base 8, because

1 times m-reduction [i] would yield (59, 125, 4935)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 77264 425823 958402 887963 853312 862437 967386 532777 551179 493601 056443 348267 630816 901456 937562 174672 613282 154470 237340 > 8¹²⁵ [i]