MinT - Best Known (239−125, 239, s)-Nets in Base 4

Best Known (239−125, 239, s)-Nets in Base 4

(239−125, 239, 130)-Net over F₄ — Constructive and digital

Digital (114, 239, 130)-net over F₄, using

t-expansion [i] based on digital (105, 239, 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]

(239−125, 239, 165)-Net over F₄ — Digital

Digital (114, 239, 165)-net over F₄, using

t-expansion [i] based on digital (109, 239, 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]

(239−125, 239, 1582)-Net in Base 4 — Upper bound on s

There is no (114, 239, 1583)-net in base 4, because

1 times m-reduction [i] would yield (114, 238, 1583)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 198640 933451 049303 600171 801753 557045 946079 008075 942414 729880 111145 210607 949017 340851 540682 064049 071215 036310 664803 669413 025432 277547 615299 811360 > 4²³⁸ [i]