MinT - Best Known (253−101, 253, s)-Nets in Base 4

Best Known (253−101, 253, s)-Nets in Base 4

(253−101, 253, 138)-Net over F₄ — Constructive and digital

Digital (152, 253, 138)-net over F₄, using

t-expansion [i] based on digital (149, 253, 138)-net over F₄, using
- 6 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]

(253−101, 253, 146)-Net in Base 4 — Constructive

(152, 253, 146)-net in base 4, using

3 times m-reduction [i] based on (152, 256, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 79, 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, 177, 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]

(253−101, 253, 363)-Net over F₄ — Digital

Digital (152, 253, 363)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−101, 253, 6988)-Net in Base 4 — Upper bound on s

There is no (152, 253, 6989)-net in base 4, because

1 times m-reduction [i] would yield (152, 252, 6989)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 599892 779167 983962 270301 678186 809793 514004 285944 852109 538310 918468 825593 910766 949441 161722 829929 320933 562103 652998 985399 065578 155286 872661 178270 621736 > 4²⁵² [i]