MinT - Best Known (111, 111+55, s)-Nets in Base 8

Best Known (111, 111+55, s)-Nets in Base 8

(111, 111+55, 1026)-Net over F₈ — Constructive and digital

Digital (111, 166, 1026)-net over F₈, using

trace code for nets [i] based on digital (28, 83, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(111, 111+55, 1816)-Net over F₈ — Digital

Digital (111, 166, 1816)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+55, 515439)-Net in Base 8 — Upper bound on s

There is no (111, 166, 515440)-net in base 8, because

1 times m-reduction [i] would yield (111, 165, 515440)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102295 602433 693562 383421 472862 543191 820076 212740 085199 831112 451110 666160 647294 518935 827300 507909 748862 299228 065896 349582 912549 677705 785732 245092 798922 > 8¹⁶⁵ [i]