MinT - Best Known (242−139, 242, s)-Nets in Base 4

Best Known (242−139, 242, s)-Nets in Base 4

(242−139, 242, 104)-Net over F₄ — Constructive and digital

Digital (103, 242, 104)-net over F₄, using

t-expansion [i] based on digital (73, 242, 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]

(242−139, 242, 144)-Net over F₄ — Digital

Digital (103, 242, 144)-net over F₄, using

t-expansion [i] based on digital (91, 242, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(242−139, 242, 1064)-Net in Base 4 — Upper bound on s

There is no (103, 242, 1065)-net in base 4, because

1 times m-reduction [i] would yield (103, 241, 1065)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 846072 668788 122744 928243 603138 993752 679195 573820 993484 526378 008716 948224 195157 054343 934658 874463 182722 482726 791490 385081 014933 756757 681649 364480 > 4²⁴¹ [i]