MinT - Best Known (247−95, 247, s)-Nets in Base 4

Best Known (247−95, 247, s)-Nets in Base 4

(247−95, 247, 138)-Net over F₄ — Constructive and digital

Digital (152, 247, 138)-net over F₄, using

t-expansion [i] based on digital (149, 247, 138)-net over F₄, using
- 12 times m-reduction [i] based on digital (149, 259, 138)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (21, 76, 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, 183, 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]

(247−95, 247, 147)-Net in Base 4 — Constructive

(152, 247, 147)-net in base 4, using

3 times m-reduction [i] based on (152, 250, 147)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (30, 79, 43)-net in base 4, using
    - net from sequence [i] based on (30, 42)-sequence in base 4, using
      - base expansion [i] based on digital (60, 42)-sequence over F₂, using
        t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [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, 171, 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]

(247−95, 247, 401)-Net over F₄ — Digital

Digital (152, 247, 401)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−95, 247, 8634)-Net in Base 4 — Upper bound on s

There is no (152, 247, 8635)-net in base 4, because

1 times m-reduction [i] would yield (152, 246, 8635)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12805 740435 710001 954898 056534 433507 800679 336484 351546 197563 096417 215240 776384 209922 438549 591413 653537 472283 616937 707198 006127 661193 008967 793601 276576 > 4²⁴⁶ [i]