MinT - Best Known (160, 160+86, s)-Nets in Base 4

Best Known (160, 160+86, s)-Nets in Base 4

(160, 160+86, 163)-Net over F₄ — Constructive and digital

Digital (160, 246, 163)-net over F₄, using

3 times m-reduction [i] based on digital (160, 249, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 59, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (101, 190, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 95, 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]

(160, 160+86, 240)-Net in Base 4 — Constructive

(160, 246, 240)-net in base 4, using

2 times m-reduction [i] based on (160, 248, 240)-net in base 4, using
- trace code for nets [i] based on (36, 124, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (36, 125, 120)-net in base 16, using
    - base change [i] based on digital (11, 100, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(160, 160+86, 558)-Net over F₄ — Digital

Digital (160, 246, 558)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(160, 160+86, 15621)-Net in Base 4 — Upper bound on s

There is no (160, 246, 15622)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12820 240754 699567 402254 616087 894976 314454 419771 577384 737947 237039 422813 750253 521250 754329 927613 091970 387381 065524 270891 279493 992703 713145 055144 468288 > 4²⁴⁶ [i]