MinT - Best Known (40, 40+33, s)-Nets in Base 81

Best Known (40, 40+33, s)-Nets in Base 81

(40, 40+33, 730)-Net over F₈₁ — Constructive and digital

Digital (40, 73, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 73, 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+33, 4188)-Net over F₈₁ — Digital

Digital (40, 73, 4188)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷³, 4188, F₈₁, 33) (dual of [4188, 4115, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁷³, 6587, F₈₁, 33) (dual of [6587, 6514, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(23) [i] based on
    1. linear OA(81⁶⁵, 6561, F₈₁, 33) (dual of [6561, 6496, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    2. linear OA(81⁴⁷, 6561, F₈₁, 24) (dual of [6561, 6514, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
    3. linear OA(81⁸, 26, F₈₁, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,81)), using
      - discarding factors / shortening the dual code based on linear OA(81⁸, 81, F₈₁, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
        Reedâ€“Solomon code RS(73,81) [i]

(40, 40+33, large)-Net in Base 81 — Upper bound on s

There is no (40, 73, large)-net in base 81, because

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