MinT - Best Known (246−135, 246, s)-Nets in Base 4

Best Known (246−135, 246, s)-Nets in Base 4

(246−135, 246, 130)-Net over F₄ — Constructive and digital

Digital (111, 246, 130)-net over F₄, using

t-expansion [i] based on digital (105, 246, 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]

(246−135, 246, 165)-Net over F₄ — Digital

Digital (111, 246, 165)-net over F₄, using

t-expansion [i] based on digital (109, 246, 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]

(246−135, 246, 1312)-Net in Base 4 — Upper bound on s

There is no (111, 246, 1313)-net in base 4, because

1 times m-reduction [i] would yield (111, 245, 1313)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3252 572617 781645 228308 234916 009187 092876 076842 605189 714536 767843 304422 501015 702484 326235 351946 895771 189189 595818 506234 844444 728660 672442 187726 179200 > 4²⁴⁵ [i]