MinT - Best Known (258−104, 258, s)-Nets in Base 4

Best Known (258−104, 258, s)-Nets in Base 4

(258−104, 258, 138)-Net over F₄ — Constructive and digital

Digital (154, 258, 138)-net over F₄, using

t-expansion [i] based on digital (149, 258, 138)-net over F₄, using
- 1 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]

(258−104, 258, 146)-Net in Base 4 — Constructive

(154, 258, 146)-net in base 4, using

t-expansion [i] based on (153, 258, 146)-net in base 4, using
- 1 times m-reduction [i] based on (153, 259, 146)-net in base 4, using
  - (u,Â u+v)-construction [i] based on
    1. (27, 80, 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, 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]

(258−104, 258, 358)-Net over F₄ — Digital

Digital (154, 258, 358)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−104, 258, 6502)-Net in Base 4 — Upper bound on s

There is no (154, 258, 6503)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 215508 912283 168708 194647 345861 469689 630380 704567 361307 227831 606820 355261 043243 304035 923231 968828 053012 769227 260208 403856 442191 676525 171749 053274 203894 445508 > 4²⁵⁸ [i]