MinT - Best Known (242−94, 242, s)-Nets in Base 4

Best Known (242−94, 242, s)-Nets in Base 4

(242−94, 242, 138)-Net over F₄ — Constructive and digital

Digital (148, 242, 138)-net over F₄, using

14 times m-reduction [i] based on digital (148, 256, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 75, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 181, 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]

(242−94, 242, 146)-Net in Base 4 — Constructive

(148, 242, 146)-net in base 4, using

2 times m-reduction [i] based on (148, 244, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 75, 42)-net in base 4, using
    - net from sequence [i] based on (27, 41)-sequence in base 4, using
      - base expansion [i] based on digital (54, 41)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [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]

(242−94, 242, 381)-Net over F₄ — Digital

Digital (148, 242, 381)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−94, 242, 7669)-Net in Base 4 — Upper bound on s

There is no (148, 242, 7670)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 102667 344121 998356 644252 472543 292689 615928 470875 530670 122388 773757 992935 982455 825634 429056 471195 430368 145207 636777 555027 625370 318083 320675 632648 > 4²⁴² [i]