MinT - Best Known (253−137, 253, s)-Nets in Base 4

Best Known (253−137, 253, s)-Nets in Base 4

(253−137, 253, 130)-Net over F₄ — Constructive and digital

Digital (116, 253, 130)-net over F₄, using

t-expansion [i] based on digital (105, 253, 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]

(253−137, 253, 168)-Net over F₄ — Digital

Digital (116, 253, 168)-net over F₄, using

t-expansion [i] based on digital (115, 253, 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]

(253−137, 253, 1429)-Net in Base 4 — Upper bound on s

There is no (116, 253, 1430)-net in base 4, because

1 times m-reduction [i] would yield (116, 252, 1430)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 54 271958 005153 576790 274847 976280 527347 642914 672773 084255 016426 351273 722377 356454 684855 844554 581202 921411 902506 888095 951768 363666 144414 761705 035126 163562 > 4²⁵² [i]