MinT - Best Known (243−131, 243, s)-Nets in Base 4

Best Known (243−131, 243, s)-Nets in Base 4

(243−131, 243, 130)-Net over F₄ — Constructive and digital

Digital (112, 243, 130)-net over F₄, using

t-expansion [i] based on digital (105, 243, 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]

(243−131, 243, 165)-Net over F₄ — Digital

Digital (112, 243, 165)-net over F₄, using

t-expansion [i] based on digital (109, 243, 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]

(243−131, 243, 1402)-Net in Base 4 — Upper bound on s

There is no (112, 243, 1403)-net in base 4, because

1 times m-reduction [i] would yield (112, 242, 1403)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 187731 381605 415966 166460 565301 462858 793804 914469 805257 289793 201958 680851 511213 399401 743206 476218 740963 929888 485388 136892 313505 061165 270190 989630 > 4²⁴² [i]