MinT - Best Known (242−91, 242, s)-Nets in Base 4

Best Known (242−91, 242, s)-Nets in Base 4

(242−91, 242, 160)-Net over F₄ — Constructive and digital

Digital (151, 242, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 78, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 164, 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]

(242−91, 242, 425)-Net over F₄ — Digital

Digital (151, 242, 425)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−91, 242, 9812)-Net in Base 4 — Upper bound on s

There is no (151, 242, 9813)-net in base 4, because

1 times m-reduction [i] would yield (151, 241, 9813)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 507472 897321 134560 291269 850953 602351 144202 763060 837681 239660 713925 951536 442658 763804 694802 235841 361198 086550 767628 228252 183362 215311 049841 218640 > 4²⁴¹ [i]