MinT - Best Known (83−40, 83, s)-Nets in Base 4

Best Known (83−40, 83, s)-Nets in Base 4

(83−40, 83, 57)-Net over F₄ — Constructive and digital

Digital (43, 83, 57)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 30, 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, 53, 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]

(83−40, 83, 65)-Net in Base 4 — Constructive

(43, 83, 65)-net in base 4, using

4 times m-reduction [i] based on (43, 87, 65)-net in base 4, using
- base change [i] based on digital (14, 58, 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]

(83−40, 83, 78)-Net over F₄ — Digital

Digital (43, 83, 78)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(83−40, 83, 856)-Net in Base 4 — Upper bound on s

There is no (43, 83, 857)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 94 864258 188734 333334 346762 103683 196877 081351 667594 > 4⁸³ [i]