MinT - Best Known (237−127, 237, s)-Nets in Base 4

Best Known (237−127, 237, s)-Nets in Base 4

(237−127, 237, 130)-Net over F₄ — Constructive and digital

Digital (110, 237, 130)-net over F₄, using

t-expansion [i] based on digital (105, 237, 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]

(237−127, 237, 165)-Net over F₄ — Digital

Digital (110, 237, 165)-net over F₄, using

t-expansion [i] based on digital (109, 237, 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]

(237−127, 237, 1407)-Net in Base 4 — Upper bound on s

There is no (110, 237, 1408)-net in base 4, because

1 times m-reduction [i] would yield (110, 236, 1408)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12665 121353 730073 410977 676154 262290 188595 903155 654054 122257 878149 772074 820975 837074 900126 437821 849262 939642 314891 020938 925967 978687 407925 755125 > 4²³⁶ [i]