MinT - Best Known (259−113, 259, s)-Nets in Base 4

Best Known (259−113, 259, s)-Nets in Base 4

(259−113, 259, 137)-Net over F₄ — Constructive and digital

Digital (146, 259, 137)-net over F₄, using

t-expansion [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]

(259−113, 259, 278)-Net over F₄ — Digital

Digital (146, 259, 278)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−113, 259, 4252)-Net in Base 4 — Upper bound on s

There is no (146, 259, 4253)-net in base 4, because

1 times m-reduction [i] would yield (146, 258, 4253)-net in base 4, but
- 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]