MinT - Best Known (153−87, 153, s)-Nets in Base 4

Best Known (153−87, 153, s)-Nets in Base 4

(153−87, 153, 66)-Net over F₄ — Constructive and digital

Digital (66, 153, 66)-net over F₄, using

t-expansion [i] based on digital (49, 153, 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]

(153−87, 153, 99)-Net over F₄ — Digital

Digital (66, 153, 99)-net over F₄, using

t-expansion [i] based on digital (61, 153, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(153−87, 153, 721)-Net in Base 4 — Upper bound on s

There is no (66, 153, 722)-net in base 4, because

1 times m-reduction [i] would yield (66, 152, 722)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 33 821677 021486 266600 426712 433729 206944 849671 510580 853110 668453 282533 480808 029811 159957 430544 > 4¹⁵² [i]