MinT - Best Known (242−84, 242, s)-Nets in Base 4

Best Known (242−84, 242, s)-Nets in Base 4

(242−84, 242, 163)-Net over F₄ — Constructive and digital

Digital (158, 242, 163)-net over F₄, using

t-expansion [i] based on digital (157, 242, 163)-net over F₄, using
- 2 times m-reduction [i] based on digital (157, 244, 163)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 58, 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 (99, 186, 130)-net over F₄, using
      - trace code for nets [i] based on digital (6, 93, 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]

(242−84, 242, 240)-Net in Base 4 — Constructive

(158, 242, 240)-net in base 4, using

2 times m-reduction [i] based on (158, 244, 240)-net in base 4, using
- trace code for nets [i] based on (36, 122, 120)-net in base 16, using
  - 3 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]

(242−84, 242, 563)-Net over F₄ — Digital

Digital (158, 242, 563)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−84, 242, 16172)-Net in Base 4 — Upper bound on s

There is no (158, 242, 16173)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 055997 243370 545424 764504 538533 677286 451149 437148 048490 819058 765474 528908 434260 077260 432429 185757 140060 285752 490716 573254 100617 543490 592160 648546 > 4²⁴² [i]