MinT - Best Known (252−127, 252, s)-Nets in Base 4

Best Known (252−127, 252, s)-Nets in Base 4

(252−127, 252, 130)-Net over F₄ — Constructive and digital

Digital (125, 252, 130)-net over F₄, using

t-expansion [i] based on digital (105, 252, 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]

(252−127, 252, 176)-Net over F₄ — Digital

Digital (125, 252, 176)-net over F₄, using

net from sequence [i] based on digital (125, 175)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 125 and N(F) ≥ 176, using
  - Drinfeld modules of rank 1 [i]

(252−127, 252, 1977)-Net in Base 4 — Upper bound on s

There is no (125, 252, 1978)-net in base 4, because

1 times m-reduction [i] would yield (125, 251, 1978)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 393162 483062 760218 791999 106143 458084 717326 209971 320666 910730 079456 127711 332441 686686 526964 180600 310091 928281 453216 822744 247590 872998 841428 992593 529920 > 4²⁵¹ [i]