MinT - Best Known (244−90, 244, s)-Nets in Base 4

Best Known (244−90, 244, s)-Nets in Base 4

(244−90, 244, 160)-Net over F₄ — Constructive and digital

Digital (154, 244, 160)-net over F₄, using

6 times m-reduction [i] based on digital (154, 250, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 81, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 169, 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]

(244−90, 244, 196)-Net in Base 4 — Constructive

(154, 244, 196)-net in base 4, using

trace code for nets [i] based on (32, 122, 98)-net in base 16, using
- 3 times m-reduction [i] based on (32, 125, 98)-net in base 16, using
  - base change [i] based on digital (7, 100, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(244−90, 244, 458)-Net over F₄ — Digital

Digital (154, 244, 458)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(244−90, 244, 10766)-Net in Base 4 — Upper bound on s

There is no (154, 244, 10767)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 801 424976 666134 648095 225794 763774 320037 544569 666611 239133 940110 094400 884581 647174 526669 667543 285361 245484 936720 125278 053656 909641 582155 195575 887776 > 4²⁴⁴ [i]