MinT - Best Known (242−95, 242, s)-Nets in Base 4

Best Known (242−95, 242, s)-Nets in Base 4

(242−95, 242, 138)-Net over F₄ — Constructive and digital

Digital (147, 242, 138)-net over F₄, using

11 times m-reduction [i] based on digital (147, 253, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 74, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 179, 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]

(242−95, 242, 146)-Net in Base 4 — Constructive

(147, 242, 146)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (27, 74, 42)-net in base 4, using
  - net from sequence [i] based on (27, 41)-sequence in base 4, using
    - base expansion [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]
2. digital (73, 168, 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]

(242−95, 242, 368)-Net over F₄ — Digital

Digital (147, 242, 368)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−95, 242, 7445)-Net in Base 4 — Upper bound on s

There is no (147, 242, 7446)-net in base 4, because

1 times m-reduction [i] would yield (147, 241, 7446)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 529369 281025 265207 830204 786157 922038 691827 060534 861963 570439 402020 851970 166342 497902 227292 716843 863247 784292 263615 623511 849466 511412 955892 080320 > 4²⁴¹ [i]