MinT - Best Known (257−147, 257, s)-Nets in Base 4

Best Known (257−147, 257, s)-Nets in Base 4

(257−147, 257, 130)-Net over F₄ — Constructive and digital

Digital (110, 257, 130)-net over F₄, using

t-expansion [i] based on digital (105, 257, 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]

(257−147, 257, 165)-Net over F₄ — Digital

Digital (110, 257, 165)-net over F₄, using

t-expansion [i] based on digital (109, 257, 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]

(257−147, 257, 1147)-Net in Base 4 — Upper bound on s

There is no (110, 257, 1148)-net in base 4, because

1 times m-reduction [i] would yield (110, 256, 1148)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14116 716130 001102 990711 198711 399267 785799 898995 689004 265407 554981 532143 765002 306403 298130 641393 614848 425288 079710 640744 643332 880555 501999 102116 359036 792065 > 4²⁵⁶ [i]