MinT - Best Known (72−33, 72, s)-Nets in Base 81

Best Known (72−33, 72, s)-Nets in Base 81

(72−33, 72, 730)-Net over F₈₁ — Constructive and digital

Digital (39, 72, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 72, 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]

(72−33, 72, 3633)-Net over F₈₁ — Digital

Digital (39, 72, 3633)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷², 3633, F₈₁, 33) (dual of [3633, 3561, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁷², 6585, F₈₁, 33) (dual of [6585, 6513, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,12]) [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(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(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]

(72−33, 72, large)-Net in Base 81 — Upper bound on s

There is no (39, 72, large)-net in base 81, because

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