MinT - Best Known (18, 18+86, s)-Nets in Base 7

Best Known (18, 18+86, s)-Nets in Base 7

(18, 18+86, 26)-Net over F₇ — Constructive and digital

Digital (18, 104, 26)-net over F₇, using

net from sequence [i] based on digital (18, 25)-sequence over F₇, using
- Niederreiter sequence [i]

(18, 18+86, 51)-Net over F₇ — Digital

Digital (18, 104, 51)-net over F₇, using

net from sequence [i] based on digital (18, 50)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 18 and N(F) ≥ 51, using
  - an algebraic function field by Teo [i]

(18, 18+86, 256)-Net over F₇ — Upper bound on s (digital)

There is no digital (18, 104, 257)-net over F₇, because

2 times m-reduction [i] would yield digital (18, 102, 257)-net over F₇, but
- extracting embedded orthogonal array [i] would yield linear OA(7¹⁰², 257, F₇, 84) (dual of [257, 155, 85]-code), but
  - residual code [i] would yield OA(7¹⁸, 172, S₇, 12), but
    - the linear programming bound shows that MÂ â‰¥ 4 737429 508718 418109 748150 / 2895 922947 > 7¹⁸ [i]

(18, 18+86, 283)-Net in Base 7 — Upper bound on s

There is no (18, 104, 284)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 7784 305516 218899 789843 494286 348580 952280 589159 302170 405575 593919 170509 593011 685503 099425 > 7¹⁰⁴ [i]