MinT - Best Known (221−76, 221, s)-Nets in Base 4

Best Known (221−76, 221, s)-Nets in Base 4

(221−76, 221, 163)-Net over F₄ — Constructive and digital

Digital (145, 221, 163)-net over F₄, using

3 times m-reduction [i] based on digital (145, 224, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 54, 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 (91, 170, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 85, 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]

(221−76, 221, 240)-Net in Base 4 — Constructive

(145, 221, 240)-net in base 4, using

4¹ times duplication [i] based on (144, 220, 240)-net in base 4, using
- t-expansion [i] based on (143, 220, 240)-net in base 4, using
  - trace code for nets [i] based on (33, 110, 120)-net in base 16, using
    - base change [i] based on digital (11, 88, 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]

(221−76, 221, 537)-Net over F₄ — Digital

Digital (145, 221, 537)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(221−76, 221, 15859)-Net in Base 4 — Upper bound on s

There is no (145, 221, 15860)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 375817 719196 299957 152441 300714 087645 935226 558624 612878 520556 208517 064832 203094 358857 567328 530636 897111 241735 810718 097444 579280 739740 > 4²²¹ [i]