MinT - Best Known (177−62, 177, s)-Nets in Base 4

Best Known (177−62, 177, s)-Nets in Base 4

(177−62, 177, 157)-Net over F₄ — Constructive and digital

Digital (115, 177, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 41, 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 (74, 136, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 68, 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]

(177−62, 177, 196)-Net in Base 4 — Constructive

(115, 177, 196)-net in base 4, using

3 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]

(177−62, 177, 414)-Net over F₄ — Digital

Digital (115, 177, 414)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(177−62, 177, 11311)-Net in Base 4 — Upper bound on s

There is no (115, 177, 11312)-net in base 4, because

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]