MinT - Best Known (23, 23+26, s)-Nets in Base 81

Best Known (23, 23+26, s)-Nets in Base 81

(23, 23+26, 370)-Net over F₈₁ — Constructive and digital

Digital (23, 49, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 49, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(23, 23+26, 793)-Net over F₈₁ — Digital

Digital (23, 49, 793)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁴⁹, 793, F₈₁, 26) (dual of [793, 744, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁴⁹, 820, F₈₁, 26) (dual of [820, 771, 27]-code), using
  - the BCH-code C(I) with length 820Â | 81²âˆ’1, defining interval IÂ =Â {1,18,35,…,426}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(23, 23+26, 1106201)-Net in Base 81 — Upper bound on s

There is no (23, 49, 1106202)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 3279 205833 173671 072419 170998 143097 095508 194834 821021 697418 788321 928620 587992 177111 828099 275681 > 81⁴⁹ [i]