MinT - Best Known (258−117, 258, s)-Nets in Base 4

Best Known (258−117, 258, s)-Nets in Base 4

(258−117, 258, 131)-Net over F₄ — Constructive and digital

Digital (141, 258, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 68, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 190, 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]

(258−117, 258, 245)-Net over F₄ — Digital

Digital (141, 258, 245)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−117, 258, 3434)-Net in Base 4 — Upper bound on s

There is no (141, 258, 3435)-net in base 4, because

1 times m-reduction [i] would yield (141, 257, 3435)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53970 200429 272320 557103 341565 019696 907220 143611 483923 687606 286459 229272 853397 700575 886022 979320 892294 904899 769401 433843 707644 197394 669918 244549 769627 056916 > 4²⁵⁷ [i]