MinT - Best Known (244−135, 244, s)-Nets in Base 4

Best Known (244−135, 244, s)-Nets in Base 4

(244−135, 244, 130)-Net over F₄ — Constructive and digital

Digital (109, 244, 130)-net over F₄, using

t-expansion [i] based on digital (105, 244, 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]

(244−135, 244, 165)-Net over F₄ — Digital

Digital (109, 244, 165)-net over F₄, using

net from sequence [i] based on digital (109, 164)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
  - Drinfeld modules of rank 1 [i]

(244−135, 244, 1257)-Net in Base 4 — Upper bound on s

There is no (109, 244, 1258)-net in base 4, because

1 times m-reduction [i] would yield (109, 243, 1258)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 207 484169 802142 176818 477513 252139 650663 427248 676999 446918 097108 433431 830038 840987 145989 960176 785947 403746 559619 652735 020410 066832 045075 712324 638604 > 4²⁴³ [i]