MinT - Best Known (245−104, 245, s)-Nets in Base 4

Best Known (245−104, 245, s)-Nets in Base 4

(245−104, 245, 137)-Net over F₄ — Constructive and digital

Digital (141, 245, 137)-net over F₄, using

2 times m-reduction [i] based on digital (141, 247, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 68, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, 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]

(245−104, 245, 289)-Net over F₄ — Digital

Digital (141, 245, 289)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−104, 245, 4585)-Net in Base 4 — Upper bound on s

There is no (141, 245, 4586)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3216 924864 242473 093211 253460 592282 241006 253149 585197 414951 081315 540024 109218 409105 263953 978873 108244 275786 959320 034187 917970 440934 387185 465160 264120 > 4²⁴⁵ [i]