MinT - Best Known (256−139, 256, s)-Nets in Base 4

Best Known (256−139, 256, s)-Nets in Base 4

(256−139, 256, 130)-Net over F₄ — Constructive and digital

Digital (117, 256, 130)-net over F₄, using

t-expansion [i] based on digital (105, 256, 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]

(256−139, 256, 168)-Net over F₄ — Digital

Digital (117, 256, 168)-net over F₄, using

t-expansion [i] based on digital (115, 256, 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]

(256−139, 256, 1428)-Net in Base 4 — Upper bound on s

There is no (117, 256, 1429)-net in base 4, because

1 times m-reduction [i] would yield (117, 255, 1429)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3474 119634 597288 084421 913651 120621 351884 299093 293160 942116 573927 475375 026901 814249 412482 598412 063794 316013 392802 526241 606563 430056 396574 239023 411186 073608 > 4²⁵⁵ [i]