MinT - Best Known (248−127, 248, s)-Nets in Base 4

Best Known (248−127, 248, s)-Nets in Base 4

(248−127, 248, 130)-Net over F₄ — Constructive and digital

Digital (121, 248, 130)-net over F₄, using

t-expansion [i] based on digital (105, 248, 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]

(248−127, 248, 168)-Net over F₄ — Digital

Digital (121, 248, 168)-net over F₄, using

t-expansion [i] based on digital (115, 248, 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]

(248−127, 248, 1806)-Net in Base 4 — Upper bound on s

There is no (121, 248, 1807)-net in base 4, because

1 times m-reduction [i] would yield (121, 247, 1807)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52232 538858 146408 531640 086267 977279 135939 405553 586416 091121 743338 084705 645501 376826 652096 651176 418651 437845 239011 992019 423967 272593 136801 672711 161280 > 4²⁴⁷ [i]