MinT - Best Known (30, 63, s)-Nets in Base 81

Best Known (30, 63, s)-Nets in Base 81

(30, 63, 370)-Net over F₈₁ — Constructive and digital

Digital (30, 63, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 63, 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]

(30, 63, 1004)-Net over F₈₁ — Digital

Digital (30, 63, 1004)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁶³, 1004, F₈₁, 33) (dual of [1004, 941, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁶³, 1312, F₈₁, 33) (dual of [1312, 1249, 34]-code), using
  - the BCH-code C(I) with length 1312Â | 81²âˆ’1, defining interval IÂ =Â {2,7,12,…,162}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(30, 63, 2112645)-Net in Base 81 — Upper bound on s

There is no (30, 63, 2112646)-net in base 81, because

1 times m-reduction [i] would yield (30, 62, 2112646)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 21187 183686 238568 046472 220778 302428 043963 742767 191298 914720 198961 219832 437471 924624 758011 032473 399352 281742 425145 914881 > 81⁶² [i]