MinT - Best Known (259−104, 259, s)-Nets in Base 4

Best Known (259−104, 259, s)-Nets in Base 4

(259−104, 259, 138)-Net over F₄ — Constructive and digital

Digital (155, 259, 138)-net over F₄, using

t-expansion [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]

(259−104, 259, 147)-Net in Base 4 — Constructive

(155, 259, 147)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (30, 82, 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, 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]

(259−104, 259, 363)-Net over F₄ — Digital

Digital (155, 259, 363)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−104, 259, 6679)-Net in Base 4 — Upper bound on s

There is no (155, 259, 6680)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 863036 638836 239892 139329 686955 749734 083134 095451 372364 530881 645746 346286 407734 186902 474409 533406 151559 105369 848921 310921 611063 984357 943474 043588 148734 068848 > 4²⁵⁹ [i]