MinT - Best Known (243−89, 243, s)-Nets in Base 4

Best Known (243−89, 243, s)-Nets in Base 4

(243−89, 243, 160)-Net over F₄ — Constructive and digital

Digital (154, 243, 160)-net over F₄, using

7 times m-reduction [i] based on digital (154, 250, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 81, 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, 169, 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]

(243−89, 243, 196)-Net in Base 4 — Constructive

(154, 243, 196)-net in base 4, using

1 times m-reduction [i] based on (154, 244, 196)-net in base 4, using
- trace code for nets [i] based on (32, 122, 98)-net in base 16, using
  - 3 times m-reduction [i] based on (32, 125, 98)-net in base 16, using
    - base change [i] based on digital (7, 100, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(243−89, 243, 468)-Net over F₄ — Digital

Digital (154, 243, 468)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−89, 243, 11743)-Net in Base 4 — Upper bound on s

There is no (154, 243, 11744)-net in base 4, because

1 times m-reduction [i] would yield (154, 242, 11744)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 041476 315747 668774 866695 995487 964761 640111 143692 171289 088193 183718 355397 173086 929494 286541 897383 661344 935351 917191 884285 387848 805195 438440 881998 > 4²⁴² [i]