MinT - Best Known (257−143, 257, s)-Nets in Base 4

Best Known (257−143, 257, s)-Nets in Base 4

(257−143, 257, 130)-Net over F₄ — Constructive and digital

Digital (114, 257, 130)-net over F₄, using

t-expansion [i] based on digital (105, 257, 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]

(257−143, 257, 165)-Net over F₄ — Digital

Digital (114, 257, 165)-net over F₄, using

t-expansion [i] based on digital (109, 257, 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]

(257−143, 257, 1289)-Net in Base 4 — Upper bound on s

There is no (114, 257, 1290)-net in base 4, because

1 times m-reduction [i] would yield (114, 256, 1290)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14040 381019 472543 101690 764196 705644 516731 246984 618097 977742 071933 259534 028712 927964 897970 683352 608572 332268 541609 573672 611154 273902 318018 092677 960366 368080 > 4²⁵⁶ [i]