MinT - Best Known (243−83, 243, s)-Nets in Base 4

Best Known (243−83, 243, s)-Nets in Base 4

(243−83, 243, 170)-Net over F₄ — Constructive and digital

Digital (160, 243, 170)-net over F₄, using

4¹ times duplication [i] based on 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]

(243−83, 243, 240)-Net in Base 4 — Constructive

(160, 243, 240)-net in base 4, using

5 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]

(243−83, 243, 598)-Net over F₄ — Digital

Digital (160, 243, 598)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−83, 243, 19215)-Net in Base 4 — Upper bound on s

There is no (160, 243, 19216)-net in base 4, because

1 times m-reduction [i] would yield (160, 242, 19216)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49 988881 154201 819768 809355 275748 009373 374712 804408 142308 446608 783800 668286 026441 626306 222528 882921 709755 267787 974991 933175 726583 908475 114258 107544 > 4²⁴² [i]