MinT - Best Known (89, 89+157, s)-Nets in Base 3

Best Known (89, 89+157, s)-Nets in Base 3

(89, 89+157, 64)-Net over F₃ — Constructive and digital

Digital (89, 246, 64)-net over F₃, using

net from sequence [i] based on digital (89, 63)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
    - a function field by GarcÃa and Quoos [i]

(89, 89+157, 96)-Net over F₃ — Digital

Digital (89, 246, 96)-net over F₃, using

net from sequence [i] based on digital (89, 95)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(89, 89+157, 389)-Net over F₃ — Upper bound on s (digital)

There is no digital (89, 246, 390)-net over F₃, because

1 times m-reduction [i] would yield digital (89, 245, 390)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁵, 390, F₃, 156) (dual of [390, 145, 157]-code), but
  - residual code [i] would yield linear OA(3⁸⁹, 233, F₃, 52) (dual of [233, 144, 53]-code), but
    - the Johnson bound shows that NÂ â‰¤ 495 041572 388827 382337 917134 411719 483784 605757 670798 500294 630099 278193 < 3¹⁴⁴ [i]

(89, 89+157, 398)-Net in Base 3 — Upper bound on s

There is no (89, 246, 399)-net in base 3, because

1 times m-reduction [i] would yield (89, 245, 399)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 927 226067 258191 972335 912867 128033 465388 381282 952750 056386 045887 937289 803921 313487 574817 906963 822046 279965 053339 906565 > 3²⁴⁵ [i]