MinT - Best Known (178−63, 178, s)-Nets in Base 4

Best Known (178−63, 178, s)-Nets in Base 4

(178−63, 178, 152)-Net over F₄ — Constructive and digital

Digital (115, 178, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 40, 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 (75, 138, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 69, 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]

(178−63, 178, 196)-Net in Base 4 — Constructive

(115, 178, 196)-net in base 4, using

2 times m-reduction [i] based on (115, 180, 196)-net in base 4, using
- trace code for nets [i] based on (25, 90, 98)-net in base 16, using
  - base change [i] based on digital (7, 72, 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]

(178−63, 178, 401)-Net over F₄ — Digital

Digital (115, 178, 401)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−63, 178, 11311)-Net in Base 4 — Upper bound on s

There is no (115, 178, 11312)-net in base 4, because

1 times m-reduction [i] would yield (115, 177, 11312)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36774 748207 566779 183012 592888 246262 350445 717117 876930 838856 821351 677816 856570 945402 000437 440810 246250 709110 > 4¹⁷⁷ [i]