MinT - Best Known (133−103, 133, s)-Nets in Base 5

Best Known (133−103, 133, s)-Nets in Base 5

(133−103, 133, 51)-Net over F₅ — Constructive and digital

Digital (30, 133, 51)-net over F₅, using

t-expansion [i] based on digital (22, 133, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(133−103, 133, 58)-Net over F₅ — Digital

Digital (30, 133, 58)-net over F₅, using

net from sequence [i] based on digital (30, 57)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 30 and N(F) ≥ 58, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(133−103, 133, 180)-Net over F₅ — Upper bound on s (digital)

There is no digital (30, 133, 181)-net over F₅, because

1 times m-reduction [i] would yield digital (30, 132, 181)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5¹³², 181, F₅, 102) (dual of [181, 49, 103]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(5¹³¹, 147, S₅, 102), but
      - the linear programming bound shows that MÂ â‰¥ 249 236477 010921 693845 684281 987701 342304 847343 669838 056342 020625 628930 208250 721989 315934 479236 602783 203125 / 6 326951 585981 > 5¹³¹ [i]
    2. OA(5⁴⁹, 181, S₅, 34), but
      - discarding factors would yield OA(5⁴⁹, 179, S₅, 34), but
        the linear programming bound shows that MÂ â‰¥ 63101 461737 633561 195888 825311 804301 658299 627461 880998 569540 679454 803466 796875 / 3 393659 613629 526150 016067 267200 843099 664623 > 5⁴⁹ [i]

(133−103, 133, 283)-Net in Base 5 — Upper bound on s

There is no (30, 133, 284)-net in base 5, because

1 times m-reduction [i] would yield (30, 132, 284)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 197 060313 469893 377004 738724 396420 808233 467567 986656 312065 432700 141226 554109 357854 233604 982385 > 5¹³² [i]