MinT - Best Known (41, 41+35, s)-Nets in Base 81

Best Known (41, 41+35, s)-Nets in Base 81

(41, 41+35, 730)-Net over F₈₁ — Constructive and digital

Digital (41, 76, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 76, 730)-net over F₈₁, using
- net from sequence [i] based on digital (36, 729)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 36 and N(F) ≥ 730, using
    - the Hermitian function field over F₈₁ [i]

(41, 41+35, 3563)-Net over F₈₁ — Digital

Digital (41, 76, 3563)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷⁶, 3563, F₈₁, 35) (dual of [3563, 3487, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁷⁶, 6585, F₈₁, 35) (dual of [6585, 6509, 36]-code), using
  - constructionÂ X applied to C([0,17]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(81⁶⁹, 6562, F₈₁, 35) (dual of [6562, 6493, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
    2. linear OA(81⁵³, 6562, F₈₁, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    3. linear OA(81⁷, 23, F₈₁, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,81)), using
      - discarding factors / shortening the dual code based on linear OA(81⁷, 81, F₈₁, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
        Reedâ€“Solomon code RS(74,81) [i]

(41, 41+35, large)-Net in Base 81 — Upper bound on s

There is no (41, 76, large)-net in base 81, because

33 times m-reduction [i] would yield (41, 43, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]