MinT - Best Known (56, 56+65, s)-Nets in Base 8

Best Known (56, 56+65, s)-Nets in Base 8

(56, 56+65, 111)-Net over F₈ — Constructive and digital

Digital (56, 121, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 42, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (14, 79, 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]

(56, 56+65, 153)-Net over F₈ — Digital

Digital (56, 121, 153)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+65, 4430)-Net in Base 8 — Upper bound on s

There is no (56, 121, 4431)-net in base 8, because

1 times m-reduction [i] would yield (56, 120, 4431)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 364531 094962 059202 462483 524048 376520 451185 860248 241501 304826 337325 989645 496359 564853 942479 839833 726028 222870 > 8¹²⁰ [i]