MinT - Best Known (254−139, 254, s)-Nets in Base 4

Best Known (254−139, 254, s)-Nets in Base 4

(254−139, 254, 130)-Net over F₄ — Constructive and digital

Digital (115, 254, 130)-net over F₄, using

t-expansion [i] based on 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]

(254−139, 254, 168)-Net over F₄ — Digital

Digital (115, 254, 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]

(254−139, 254, 1369)-Net in Base 4 — Upper bound on s

There is no (115, 254, 1370)-net in base 4, because

1 times m-reduction [i] would yield (115, 253, 1370)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 211 553858 255039 828797 237111 324448 926115 844034 009981 873831 874928 081159 973316 751947 064768 877734 462993 811693 123659 714348 784402 469696 920560 355980 154521 587302 > 4²⁵³ [i]