MinT - Best Known (20, 20+23, s)-Nets in Base 81

Best Known (20, 20+23, s)-Nets in Base 81

(20, 20+23, 370)-Net over F₈₁ — Constructive and digital

Digital (20, 43, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 43, 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]

(20, 20+23, 703)-Net over F₈₁ — Digital

Digital (20, 43, 703)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁴³, 703, F₈₁, 23) (dual of [703, 660, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁴³, 820, F₈₁, 23) (dual of [820, 777, 24]-code), using
  - the BCH-code C(I) with length 820Â | 81²âˆ’1, defining interval IÂ =Â {1,18,35,…,375}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(20, 20+23, 1188133)-Net in Base 81 — Upper bound on s

There is no (20, 43, 1188134)-net in base 81, because

1 times m-reduction [i] would yield (20, 42, 1188134)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 143 342329 880668 072446 714915 797343 142655 176770 402974 997873 190486 299473 817259 797921 > 81⁴² [i]