MinT - Best Known (260−111, 260, s)-Nets in Base 4

Best Known (260−111, 260, s)-Nets in Base 4

(260−111, 260, 138)-Net over F₄ — Constructive and digital

Digital (149, 260, 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, 184, 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]

(260−111, 260, 299)-Net over F₄ — Digital

Digital (149, 260, 299)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(260−111, 260, 4821)-Net in Base 4 — Upper bound on s

There is no (149, 260, 4822)-net in base 4, because

1 times m-reduction [i] would yield (149, 259, 4822)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 867380 565544 787086 372806 191814 234275 998170 692021 663341 264201 018455 293299 201540 508166 662770 449825 490558 628710 069665 511363 139313 728526 091632 411218 785219 762048 > 4²⁵⁹ [i]