MinT - Best Known (163−51, 163, s)-Nets in Base 8

Best Known (163−51, 163, s)-Nets in Base 8

(163−51, 163, 1026)-Net over F₈ — Constructive and digital

Digital (112, 163, 1026)-net over F₈, using

5 times m-reduction [i] based on digital (112, 168, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 84, 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]

(163−51, 163, 2472)-Net over F₈ — Digital

Digital (112, 163, 2472)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(163−51, 163, 1034056)-Net in Base 8 — Upper bound on s

There is no (112, 163, 1034057)-net in base 8, because

1 times m-reduction [i] would yield (112, 162, 1034057)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 199 793239 348700 948196 188173 045138 224786 131019 352678 905813 162044 227815 675753 486337 034482 549626 243363 961439 186492 291406 988463 130691 553853 756872 286432 > 8¹⁶² [i]