MinT - Best Known (255−131, 255, s)-Nets in Base 4

Best Known (255−131, 255, s)-Nets in Base 4

(255−131, 255, 130)-Net over F₄ — Constructive and digital

Digital (124, 255, 130)-net over F₄, using

t-expansion [i] based on digital (105, 255, 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]

(255−131, 255, 168)-Net over F₄ — Digital

Digital (124, 255, 168)-net over F₄, using

t-expansion [i] based on digital (115, 255, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(255−131, 255, 1827)-Net in Base 4 — Upper bound on s

There is no (124, 255, 1828)-net in base 4, because

1 times m-reduction [i] would yield (124, 254, 1828)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 857 397268 687708 244346 456790 617934 345775 010123 493101 219311 264237 170186 453149 602487 326976 421886 134065 436872 893095 299545 737449 806590 136322 173545 384998 944380 > 4²⁵⁴ [i]