MinT - Best Known (40, 40+34, s)-Nets in Base 81

Best Known (40, 40+34, s)-Nets in Base 81

(40, 40+34, 730)-Net over F₈₁ — Constructive and digital

Digital (40, 74, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 74, 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]

(40, 40+34, 3595)-Net over F₈₁ — Digital

Digital (40, 74, 3595)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷⁴, 3595, F₈₁, 34) (dual of [3595, 3521, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁷⁴, 6584, F₈₁, 34) (dual of [6584, 6510, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(25) [i] based on
    1. linear OA(81⁶⁷, 6561, F₈₁, 34) (dual of [6561, 6494, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(81⁵¹, 6561, F₈₁, 26) (dual of [6561, 6510, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [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]

(40, 40+34, large)-Net in Base 81 — Upper bound on s

There is no (40, 74, large)-net in base 81, because

32 times m-reduction [i] would yield (40, 42, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]