MinT - Best Known (230−117, 230, s)-Nets in Base 4

Best Known (230−117, 230, s)-Nets in Base 4

(230−117, 230, 130)-Net over F₄ — Constructive and digital

Digital (113, 230, 130)-net over F₄, using

t-expansion [i] based on digital (105, 230, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(230−117, 230, 165)-Net over F₄ — Digital

Digital (113, 230, 165)-net over F₄, using

t-expansion [i] based on digital (109, 230, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(230−117, 230, 1735)-Net in Base 4 — Upper bound on s

There is no (113, 230, 1736)-net in base 4, because

1 times m-reduction [i] would yield (113, 229, 1736)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 747447 077424 765371 379241 289255 904236 074041 223452 363400 382059 627236 658884 976762 099130 619134 953706 228342 629425 858831 330796 227112 599725 276920 > 4²²⁹ [i]