MinT - Best Known (254−149, 254, s)-Nets in Base 4

Best Known (254−149, 254, s)-Nets in Base 4

(254−149, 254, 130)-Net over F₄ — Constructive and digital

Digital (105, 254, 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]

(254−149, 254, 144)-Net over F₄ — Digital

Digital (105, 254, 144)-net over F₄, using

t-expansion [i] based on digital (91, 254, 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]

(254−149, 254, 1022)-Net in Base 4 — Upper bound on s

There is no (105, 254, 1023)-net in base 4, because

1 times m-reduction [i] would yield (105, 253, 1023)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 222 102475 979807 969771 460608 325932 073640 174890 381113 084566 726860 421009 037432 118018 019061 412206 644796 230774 081340 884721 009954 582376 446103 075051 021101 574931 > 4²⁵³ [i]