MinT - Best Known (20, 20+22, s)-Nets in Base 81

Best Known (20, 20+22, s)-Nets in Base 81

(20, 20+22, 370)-Net over F₈₁ — Constructive and digital

Digital (20, 42, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 42, 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+22, 840)-Net over F₈₁ — Digital

Digital (20, 42, 840)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁴², 840, F₈₁, 22) (dual of [840, 798, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁴², 1312, F₈₁, 22) (dual of [1312, 1270, 23]-code), using
  - the BCH-code C(I) with length 1312Â | 81²âˆ’1, defining interval IÂ =Â {1,6,11,…,106}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(20, 20+22, 1188133)-Net in Base 81 — Upper bound on s

There is no (20, 42, 1188134)-net in base 81, because

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]