MinT - Best Known (240−123, 240, s)-Nets in Base 4

Best Known (240−123, 240, s)-Nets in Base 4

(240−123, 240, 130)-Net over F₄ — Constructive and digital

Digital (117, 240, 130)-net over F₄, using

t-expansion [i] based on digital (105, 240, 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]

(240−123, 240, 168)-Net over F₄ — Digital

Digital (117, 240, 168)-net over F₄, using

t-expansion [i] based on digital (115, 240, 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]

(240−123, 240, 1744)-Net in Base 4 — Upper bound on s

There is no (117, 240, 1745)-net in base 4, because

1 times m-reduction [i] would yield (117, 239, 1745)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 783247 788693 648031 899029 977544 501329 740403 963676 049532 102015 988530 212883 401001 728282 578645 947954 473927 007866 911767 475531 990944 326719 381185 891840 > 4²³⁹ [i]