MinT - Best Known (43, 43+35, s)-Nets in Base 81

Best Known (43, 43+35, s)-Nets in Base 81

(43, 43+35, 730)-Net over F₈₁ — Constructive and digital

Digital (43, 78, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 78, 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]

(43, 43+35, 4655)-Net over F₈₁ — Digital

Digital (43, 78, 4655)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷⁸, 4655, F₈₁, 35) (dual of [4655, 4577, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁷⁸, 6591, F₈₁, 35) (dual of [6591, 6513, 36]-code), using
  - constructionÂ X applied to C([0,17]) âŠ‚ C([0,12]) [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(81⁴⁹, 6562, F₈₁, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(81⁹, 29, F₈₁, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,81)), using
      - discarding factors / shortening the dual code based on linear OA(81⁹, 81, F₈₁, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
        Reedâ€“Solomon code RS(72,81) [i]

(43, 43+35, large)-Net in Base 81 — Upper bound on s

There is no (43, 78, large)-net in base 81, because

33 times m-reduction [i] would yield (43, 45, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]