MinT - Best Known (254−103, 254, s)-Nets in Base 4

Best Known (254−103, 254, s)-Nets in Base 4

(254−103, 254, 138)-Net over F₄ — Constructive and digital

Digital (151, 254, 138)-net over F₄, using

t-expansion [i] based on digital (149, 254, 138)-net over F₄, using
- 5 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]

(254−103, 254, 146)-Net in Base 4 — Constructive

(151, 254, 146)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (27, 78, 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, 176, 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]

(254−103, 254, 346)-Net over F₄ — Digital

Digital (151, 254, 346)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−103, 254, 6376)-Net in Base 4 — Upper bound on s

There is no (151, 254, 6377)-net in base 4, because

1 times m-reduction [i] would yield (151, 253, 6377)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 210 384398 170256 430975 341430 824623 888536 208423 305463 049582 233205 571899 211796 851575 651771 280246 267096 358952 936432 588486 148263 069743 952741 181401 961745 065952 > 4²⁵³ [i]