MinT - Best Known (70−31, 70, s)-Nets in Base 81

Best Known (70−31, 70, s)-Nets in Base 81

(70−31, 70, 730)-Net over F₈₁ — Constructive and digital

Digital (39, 70, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 70, 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]

(70−31, 70, 5055)-Net over F₈₁ — Digital

Digital (39, 70, 5055)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷⁰, 5055, F₈₁, 31) (dual of [5055, 4985, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁷⁰, 6591, F₈₁, 31) (dual of [6591, 6521, 32]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(81⁶¹, 6562, F₈₁, 31) (dual of [6562, 6501, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(81⁴¹, 6562, F₈₁, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (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]

(70−31, 70, large)-Net in Base 81 — Upper bound on s

There is no (39, 70, large)-net in base 81, because

29 times m-reduction [i] would yield (39, 41, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]