MinT - Best Known (21, 44, s)-Nets in Base 81

Best Known (21, 44, s)-Nets in Base 81

(21, 44, 370)-Net over F₈₁ — Constructive and digital

Digital (21, 44, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 44, 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]

(21, 44, 868)-Net over F₈₁ — Digital

Digital (21, 44, 868)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁴⁴, 868, F₈₁, 23) (dual of [868, 824, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁴⁴, 1312, F₈₁, 23) (dual of [1312, 1268, 24]-code), using
  - the BCH-code C(I) with length 1312Â | 81²âˆ’1, defining interval IÂ =Â {1,6,11,…,111}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(21, 44, 1771594)-Net in Base 81 — Upper bound on s

There is no (21, 44, 1771595)-net in base 81, because

1 times m-reduction [i] would yield (21, 43, 1771595)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 11610 659726 602188 276014 607481 431035 248413 062433 768639 340239 820477 177049 538413 723601 > 81⁴³ [i]