MinT - Best Known (52, 52+117, s)-Nets in Base 8

Best Known (52, 52+117, s)-Nets in Base 8

(52, 52+117, 98)-Net over F₈ — Constructive and digital

Digital (52, 169, 98)-net over F₈, using

t-expansion [i] based on digital (37, 169, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(52, 52+117, 144)-Net over F₈ — Digital

Digital (52, 169, 144)-net over F₈, using

t-expansion [i] based on digital (45, 169, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(52, 52+117, 1287)-Net in Base 8 — Upper bound on s

There is no (52, 169, 1288)-net in base 8, because

1 times m-reduction [i] would yield (52, 168, 1288)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 862004 493560 286201 190653 551394 926146 246945 884708 679281 026779 347839 338647 319355 595836 327145 823332 070917 900618 917191 275715 661587 712066 067811 199613 920352 > 8¹⁶⁸ [i]