MinT - Best Known (72, 72+141, s)-Nets in Base 4

Best Known (72, 72+141, s)-Nets in Base 4

(72, 72+141, 66)-Net over F₄ — Constructive and digital

Digital (72, 213, 66)-net over F₄, using

t-expansion [i] based on digital (49, 213, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(72, 72+141, 105)-Net over F₄ — Digital

Digital (72, 213, 105)-net over F₄, using

t-expansion [i] based on digital (70, 213, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(72, 72+141, 541)-Net in Base 4 — Upper bound on s

There is no (72, 213, 542)-net in base 4, because

1 times m-reduction [i] would yield (72, 212, 542)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 754107 327781 291198 965151 523236 531728 389151 563388 719420 822717 694737 220763 553674 349935 576331 873002 657709 971771 571943 787685 396960 > 4²¹² [i]