MinT - Best Known (255−105, 255, s)-Nets in Base 4

Best Known (255−105, 255, s)-Nets in Base 4

(255−105, 255, 138)-Net over F₄ — Constructive and digital

Digital (150, 255, 138)-net over F₄, using

t-expansion [i] based on digital (149, 255, 138)-net over F₄, using
- 4 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]

(255−105, 255, 139)-Net in Base 4 — Constructive

(150, 255, 139)-net in base 4, using

1 times m-reduction [i] based on (150, 256, 139)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (24, 77, 35)-net in base 4, using
    - net from sequence [i] based on (24, 34)-sequence in base 4, using
      - base expansion [i] based on digital (48, 34)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
  2. digital (73, 179, 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]

(255−105, 255, 331)-Net over F₄ — Digital

Digital (150, 255, 331)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(255−105, 255, 5840)-Net in Base 4 — Upper bound on s

There is no (150, 255, 5841)-net in base 4, because

1 times m-reduction [i] would yield (150, 254, 5841)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 842 441153 537591 747353 853726 384656 054282 956818 927804 924983 224133 595201 393910 468881 620509 305418 257171 563350 008846 296050 201679 097390 553892 611938 036985 841360 > 4²⁵⁴ [i]