MinT - Best Known (109, 109+109, s)-Nets in Base 4

Best Known (109, 109+109, s)-Nets in Base 4

(109, 109+109, 130)-Net over F₄ — Constructive and digital

Digital (109, 218, 130)-net over F₄, using

t-expansion [i] based on digital (105, 218, 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]

(109, 109+109, 165)-Net over F₄ — Digital

Digital (109, 218, 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]

(109, 109+109, 1791)-Net in Base 4 — Upper bound on s

There is no (109, 218, 1792)-net in base 4, because

1 times m-reduction [i] would yield (109, 217, 1792)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44826 978678 381182 813233 750098 272172 173817 263074 339939 784478 976099 470937 130092 341796 176517 265494 980715 849955 729873 211196 724627 570665 > 4²¹⁷ [i]