MinT - Best Known (68−57, 68, s)-Nets in Base 7

Best Known (68−57, 68, s)-Nets in Base 7

(68−57, 68, 19)-Net over F₇ — Constructive and digital

Digital (11, 68, 19)-net over F₇, using

net from sequence [i] based on digital (11, 18)-sequence over F₇, using
- Niederreiter sequence [i]

(68−57, 68, 38)-Net over F₇ — Digital

Digital (11, 68, 38)-net over F₇, using

t-expansion [i] based on digital (9, 68, 38)-net over F₇, using
- net from sequence [i] based on digital (9, 37)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 9 and N(F) ≥ 38, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(68−57, 68, 110)-Net in Base 7 — Upper bound on s

There is no (11, 68, 111)-net in base 7, because

extracting embedded orthogonal array [i] would yield OA(7⁶⁸, 111, S₇, 57), but
- the linear programming bound shows that MÂ â‰¥ 65 058686 062896 567004 454153 517531 530259 352220 631855 594388 557308 268620 523281 295598 554486 605416 185981 837930 509122 801605 266951 / 20204 234956 576256 697888 532606 253882 786502 745819 122370 129902 264690 > 7⁶⁸ [i]