MinT - Best Known (121, 121+67, s)-Nets in Base 4

Best Known (121, 121+67, s)-Nets in Base 4

(121, 121+67, 152)-Net over F₄ — Constructive and digital

Digital (121, 188, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 42, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, 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]

(121, 121+67, 196)-Net in Base 4 — Constructive

(121, 188, 196)-net in base 4, using

2 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]

(121, 121+67, 411)-Net over F₄ — Digital

Digital (121, 188, 411)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(121, 121+67, 11294)-Net in Base 4 — Upper bound on s

There is no (121, 188, 11295)-net in base 4, because

1 times m-reduction [i] would yield (121, 187, 11295)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 38525 512241 542878 188692 039476 218283 555630 685378 112630 893292 353460 787936 496582 081001 043556 754214 089544 703989 887296 > 4¹⁸⁷ [i]