MinT - Best Known (139, 139+103, s)-Nets in Base 4

Best Known (139, 139+103, s)-Nets in Base 4

(139, 139+103, 137)-Net over F₄ — Constructive and digital

Digital (139, 242, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 66, 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, 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]

(139, 139+103, 283)-Net over F₄ — Digital

Digital (139, 242, 283)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+103, 4590)-Net in Base 4 — Upper bound on s

There is no (139, 242, 4591)-net in base 4, because

1 times m-reduction [i] would yield (139, 241, 4591)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 613136 798901 671562 943794 998960 545423 751289 343699 656807 505942 465141 027913 464278 895273 833276 142277 137393 350267 959501 743494 617210 820856 356474 103840 > 4²⁴¹ [i]