MinT - Best Known (258−112, 258, s)-Nets in Base 4

Best Known (258−112, 258, s)-Nets in Base 4

(258−112, 258, 137)-Net over F₄ — Constructive and digital

Digital (146, 258, 137)-net over F₄, using

t-expansion [i] based on digital (145, 258, 137)-net over F₄, using
- 1 times m-reduction [i] based on digital (145, 259, 137)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 72, 33)-net over F₄, using
      - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
    2. digital (73, 187, 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]

(258−112, 258, 281)-Net over F₄ — Digital

Digital (146, 258, 281)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−112, 258, 4252)-Net in Base 4 — Upper bound on s

There is no (146, 258, 4253)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 216458 655276 707544 637434 049028 587504 876909 511948 195478 440930 338986 600721 753327 457669 107780 929026 888025 454354 764075 223254 069344 988526 615330 662333 463482 542620 > 4²⁵⁸ [i]