MinT - Best Known (91, 91+47, s)-Nets in Base 4

Best Known (91, 91+47, s)-Nets in Base 4

(91, 91+47, 152)-Net over F₄ — Constructive and digital

Digital (91, 138, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 32, 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 (59, 106, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 53, 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]

(91, 91+47, 196)-Net in Base 4 — Constructive

(91, 138, 196)-net in base 4, using

2 times m-reduction [i] based on (91, 140, 196)-net in base 4, using
- trace code for nets [i] based on (21, 70, 98)-net in base 16, using
  - base change [i] based on digital (7, 56, 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]

(91, 91+47, 371)-Net over F₄ — Digital

Digital (91, 138, 371)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(91, 91+47, 12101)-Net in Base 4 — Upper bound on s

There is no (91, 138, 12102)-net in base 4, because

1 times m-reduction [i] would yield (91, 137, 12102)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 30366 516506 973511 727651 840943 429470 699022 168191 074021 072581 591576 794046 655908 428128 > 4¹³⁷ [i]