MinT - Best Known (129−92, 129, s)-Nets in Base 4

Best Known (129−92, 129, s)-Nets in Base 4

(129−92, 129, 56)-Net over F₄ — Constructive and digital

Digital (37, 129, 56)-net over F₄, using

t-expansion [i] based on digital (33, 129, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(129−92, 129, 66)-Net over F₄ — Digital

Digital (37, 129, 66)-net over F₄, using

net from sequence [i] based on digital (37, 65)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
  - Drinfeld modules of rank 1 [i]

(129−92, 129, 250)-Net over F₄ — Upper bound on s (digital)

There is no digital (37, 129, 251)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹²⁹, 251, F₄, 92) (dual of [251, 122, 93]-code), but
- residual code [i] would yield OA(4³⁷, 158, S₄, 23), but
  - 1 times truncation [i] would yield OA(4³⁶, 157, S₄, 22), but
    - the linear programming bound shows that MÂ â‰¥ 1 085711 389750 577086 675646 991691 769970 688000 / 226 068152 003046 919267 > 4³⁶ [i]

(129−92, 129, 256)-Net in Base 4 — Upper bound on s

There is no (37, 129, 257)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 487937 928760 194316 432188 733755 137630 116184 000836 067841 961763 943771 901087 453312 > 4¹²⁹ [i]