MinT - Best Known (116−72, 116, s)-Nets in Base 3

Best Known (116−72, 116, s)-Nets in Base 3

(116−72, 116, 42)-Net over F₃ — Constructive and digital

Digital (44, 116, 42)-net over F₃, using

t-expansion [i] based on digital (39, 116, 42)-net over F₃, using
- net from sequence [i] based on digital (39, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(116−72, 116, 56)-Net over F₃ — Digital

Digital (44, 116, 56)-net over F₃, using

t-expansion [i] based on digital (40, 116, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(116−72, 116, 196)-Net over F₃ — Upper bound on s (digital)

There is no digital (44, 116, 197)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹¹⁶, 197, F₃, 72) (dual of [197, 81, 73]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3¹¹⁵, 148, S₃, 72), but
    - the linear programming bound shows that MÂ â‰¥ 15138 532736 321222 240484 923894 686292 145850 190102 104613 028724 231594 313752 692357 / 1981 901463 555581 678825 > 3¹¹⁵ [i]
  2. OA(3⁸¹, 197, S₃, 49), but
    - discarding factors would yield OA(3⁸¹, 145, S₃, 49), but
      - the linear programming bound shows that MÂ â‰¥ 5454 903125 674302 038138 022709 253118 870603 680604 996309 484668 631548 163854 828465 899522 845238 627953 731491 227462 708997 548509 054594 512076 840977 032054 473727 036116 069732 583516 742090 980902 781249 742853 019393 892253 960097 248495 972677 421373 426141 628074 366647 / 12 085062 732503 338992 176763 549129 519718 479110 268997 150451 737221 775723 050593 147904 271161 753240 445212 100862 734331 956180 467283 411940 172483 857447 041584 506698 884680 282731 811631 103778 314212 762058 997686 304963 565740 > 3⁸¹ [i]

(116−72, 116, 212)-Net in Base 3 — Upper bound on s

There is no (44, 116, 213)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 23 614732 588038 777520 958746 415339 936255 574999 777522 410321 > 3¹¹⁶ [i]