MinT - Best Known (239−131, 239, s)-Nets in Base 4

Best Known (239−131, 239, s)-Nets in Base 4

(239−131, 239, 130)-Net over F₄ — Constructive and digital

Digital (108, 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]

(239−131, 239, 144)-Net over F₄ — Digital

Digital (108, 239, 144)-net over F₄, using

t-expansion [i] based on digital (91, 239, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(239−131, 239, 1283)-Net in Base 4 — Upper bound on s

There is no (108, 239, 1284)-net in base 4, because

1 times m-reduction [i] would yield (108, 238, 1284)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195955 104221 036519 501483 471192 425703 511939 074262 719223 489697 363974 150970 304650 898429 991928 817736 195431 422973 183453 729768 751048 969692 823725 609500 > 4²³⁸ [i]