MinT - Best Known (189−67, 189, s)-Nets in Base 4

Best Known (189−67, 189, s)-Nets in Base 4

(189−67, 189, 157)-Net over F₄ — Constructive and digital

Digital (122, 189, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 43, 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 (79, 146, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 73, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(189−67, 189, 196)-Net in Base 4 — Constructive

(122, 189, 196)-net in base 4, using

t-expansion [i] based on (121, 189, 196)-net in base 4, using
- 1 times m-reduction [i] based on (121, 190, 196)-net in base 4, using
  - trace code for nets [i] based on (26, 95, 98)-net in base 16, using
    - base change [i] based on digital (7, 76, 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]

(189−67, 189, 420)-Net over F₄ — Digital

Digital (122, 189, 420)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−67, 189, 11780)-Net in Base 4 — Upper bound on s

There is no (122, 189, 11781)-net in base 4, because

1 times m-reduction [i] would yield (122, 188, 11781)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 154207 799788 611474 033008 387437 251398 778497 608242 420712 039612 096349 308398 706009 966668 579528 806740 618740 323495 324416 > 4¹⁸⁸ [i]