MinT - Best Known (87−42, 87, s)-Nets in Base 4

Best Known (87−42, 87, s)-Nets in Base 4

(87−42, 87, 57)-Net over F₄ — Constructive and digital

Digital (45, 87, 57)-net over F₄, using

2 times m-reduction [i] based on digital (45, 89, 57)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 32, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (13, 57, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(87−42, 87, 65)-Net in Base 4 — Constructive

(45, 87, 65)-net in base 4, using

6 times m-reduction [i] based on (45, 93, 65)-net in base 4, using
- base change [i] based on digital (14, 62, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(87−42, 87, 81)-Net over F₄ — Digital

Digital (45, 87, 81)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−42, 87, 885)-Net in Base 4 — Upper bound on s

There is no (45, 87, 886)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 23955 650695 471348 009345 043254 289852 765496 228427 680229 > 4⁸⁷ [i]