MinT - Best Known (207−127, 207, s)-Nets in Base 4

Best Known (207−127, 207, s)-Nets in Base 4

(207−127, 207, 104)-Net over F₄ — Constructive and digital

Digital (80, 207, 104)-net over F₄, using

t-expansion [i] based on digital (73, 207, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(207−127, 207, 112)-Net over F₄ — Digital

Digital (80, 207, 112)-net over F₄, using

t-expansion [i] based on digital (73, 207, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(207−127, 207, 703)-Net in Base 4 — Upper bound on s

There is no (80, 207, 704)-net in base 4, because

1 times m-reduction [i] would yield (80, 206, 704)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11492 984507 568111 244478 241503 856693 188449 696911 616242 584804 804125 604627 781117 295402 032115 995641 520482 542223 710613 814484 720075 > 4²⁰⁶ [i]