MinT - Best Known (155, 242, s)-Nets in Base 4

Best Known (155, 242, s)-Nets in Base 4

(155, 242, 160)-Net over F₄ — Constructive and digital

Digital (155, 242, 160)-net over F₄, using

11 times m-reduction [i] based on digital (155, 253, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 82, 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, 171, 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]

(155, 242, 208)-Net in Base 4 — Constructive

(155, 242, 208)-net in base 4, using

4² times duplication [i] based on (153, 240, 208)-net in base 4, using
- trace code for nets [i] based on (33, 120, 104)-net in base 16, using
  - base change [i] based on digital (9, 96, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(155, 242, 498)-Net over F₄ — Digital

Digital (155, 242, 498)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(155, 242, 13290)-Net in Base 4 — Upper bound on s

There is no (155, 242, 13291)-net in base 4, because

1 times m-reduction [i] would yield (155, 241, 13291)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 519182 090367 598199 247098 350759 764222 986619 005415 145856 822413 102466 180403 341332 613030 809038 557190 428935 674278 633381 835259 376808 337215 304107 971236 > 4²⁴¹ [i]