MinT - Best Known (244−133, 244, s)-Nets in Base 4

Best Known (244−133, 244, s)-Nets in Base 4

(244−133, 244, 130)-Net over F₄ — Constructive and digital

Digital (111, 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]

(244−133, 244, 165)-Net over F₄ — Digital

Digital (111, 244, 165)-net over F₄, using

t-expansion [i] based on 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]

(244−133, 244, 1341)-Net in Base 4 — Upper bound on s

There is no (111, 244, 1342)-net in base 4, because

1 times m-reduction [i] would yield (111, 243, 1342)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204 400516 029960 102982 418060 988296 975501 703986 198968 788317 122546 112554 466018 306568 760673 736348 090627 935739 884382 378659 080311 953140 815829 371705 245080 > 4²⁴³ [i]