MinT - Best Known (236−91, 236, s)-Nets in Base 4

Best Known (236−91, 236, s)-Nets in Base 4

(236−91, 236, 138)-Net over F₄ — Constructive and digital

Digital (145, 236, 138)-net over F₄, using

11 times m-reduction [i] based on digital (145, 247, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 72, 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, 175, 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]

(236−91, 236, 146)-Net in Base 4 — Constructive

(145, 236, 146)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (27, 72, 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, 164, 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]

(236−91, 236, 382)-Net over F₄ — Digital

Digital (145, 236, 382)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(236−91, 236, 8150)-Net in Base 4 — Upper bound on s

There is no (145, 236, 8151)-net in base 4, because

1 times m-reduction [i] would yield (145, 235, 8151)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3058 402623 671267 681477 833754 731365 723379 340804 773772 018470 426181 238053 903479 340021 042494 579379 252033 176186 311136 345454 479711 122310 358917 643512 > 4²³⁵ [i]