MinT - Best Known (257−149, 257, s)-Nets in Base 4

Best Known (257−149, 257, s)-Nets in Base 4

(257−149, 257, 130)-Net over F₄ — Constructive and digital

Digital (108, 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]

(257−149, 257, 144)-Net over F₄ — Digital

Digital (108, 257, 144)-net over F₄, using

t-expansion [i] based on digital (91, 257, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(257−149, 257, 1084)-Net in Base 4 — Upper bound on s

There is no (108, 257, 1085)-net in base 4, because

1 times m-reduction [i] would yield (108, 256, 1085)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13731 927238 867155 430792 166365 120961 281260 928491 119395 635845 677828 721525 748102 315170 280689 238305 750477 160984 296556 412280 416478 432430 708939 186806 063026 674400 > 4²⁵⁶ [i]