MinT - Best Known (140, 140+105, s)-Nets in Base 4

Best Known (140, 140+105, s)-Nets in Base 4

(140, 140+105, 137)-Net over F₄ — Constructive and digital

Digital (140, 245, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 67, 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, 178, 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]

(140, 140+105, 280)-Net over F₄ — Digital

Digital (140, 245, 280)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(140, 140+105, 4463)-Net in Base 4 — Upper bound on s

There is no (140, 245, 4464)-net in base 4, because

1 times m-reduction [i] would yield (140, 244, 4464)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 802 044066 106994 915990 845796 792716 836402 776683 828937 437366 568702 759972 693905 979015 957447 930069 092437 194882 764944 334186 949364 105827 720081 552536 426900 > 4²⁴⁴ [i]