MinT - Best Known (242−83, 242, s)-Nets in Base 4

Best Known (242−83, 242, s)-Nets in Base 4

(242−83, 242, 170)-Net over F₄ — Constructive and digital

Digital (159, 242, 170)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (45, 86, 66)-net over F₄, using
  - trace code for nets [i] based on digital (2, 43, 33)-net over F₁₆, using
    - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
2. digital (73, 156, 104)-net over F₄, using
  - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(242−83, 242, 240)-Net in Base 4 — Constructive

(159, 242, 240)-net in base 4, using

4 times m-reduction [i] based on (159, 246, 240)-net in base 4, using
- trace code for nets [i] based on (36, 123, 120)-net in base 16, using
  - 2 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]

(242−83, 242, 587)-Net over F₄ — Digital

Digital (159, 242, 587)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−83, 242, 18575)-Net in Base 4 — Upper bound on s

There is no (159, 242, 18576)-net in base 4, because

1 times m-reduction [i] would yield (159, 241, 18576)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 496907 781927 891936 197058 441247 480292 436034 197321 170621 917613 345800 302141 045941 650834 963313 582073 295160 508262 837751 353514 775581 530076 093775 153028 > 4²⁴¹ [i]