MinT - Best Known (111, 111+131, s)-Nets in Base 4

Best Known (111, 111+131, s)-Nets in Base 4

(111, 111+131, 130)-Net over F₄ — Constructive and digital

Digital (111, 242, 130)-net over F₄, using

t-expansion [i] based on digital (105, 242, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(111, 111+131, 165)-Net over F₄ — Digital

Digital (111, 242, 165)-net over F₄, using

t-expansion [i] based on digital (109, 242, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(111, 111+131, 1372)-Net in Base 4 — Upper bound on s

There is no (111, 242, 1373)-net in base 4, because

1 times m-reduction [i] would yield (111, 241, 1373)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 958185 927021 071270 866230 194221 485885 774630 543671 003005 867551 334033 683335 981763 223857 753158 086105 923008 214691 074189 417119 299729 435427 851349 908696 > 4²⁴¹ [i]