MinT - Best Known (43, 43+33, s)-Nets in Base 81

Best Known (43, 43+33, s)-Nets in Base 81

(43, 43+33, 730)-Net over F₈₁ — Constructive and digital

Digital (43, 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]

(43, 43+33, 6416)-Net over F₈₁ — Digital

Digital (43, 76, 6416)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷⁶, 6416, F₈₁, 33) (dual of [6416, 6340, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁷⁶, 6597, F₈₁, 33) (dual of [6597, 6521, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(81⁶⁵, 6562, F₈₁, 33) (dual of [6562, 6497, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (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(81¹¹, 35, F₈₁, 11) (dual of [35, 24, 12]-code or 35-arc in PG(10,81)), using
      - discarding factors / shortening the dual code based on linear OA(81¹¹, 81, F₈₁, 11) (dual of [81, 70, 12]-code or 81-arc in PG(10,81)), using
        Reedâ€“Solomon code RS(70,81) [i]

(43, 43+33, large)-Net in Base 81 — Upper bound on s

There is no (43, 76, large)-net in base 81, because

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