MinT - Best Known (40, 40+31, s)-Nets in Base 81

Best Known (40, 40+31, s)-Nets in Base 81

(40, 40+31, 730)-Net over F₈₁ — Constructive and digital

Digital (40, 71, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 71, 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+31, 5884)-Net over F₈₁ — Digital

Digital (40, 71, 5884)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷¹, 5884, F₈₁, 31) (dual of [5884, 5813, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁷¹, 6593, F₈₁, 31) (dual of [6593, 6522, 32]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(19) [i] based on
    1. linear OA(81⁶¹, 6561, F₈₁, 31) (dual of [6561, 6500, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(81³⁹, 6561, F₈₁, 20) (dual of [6561, 6522, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    3. linear OA(81¹⁰, 32, F₈₁, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,81)), using
      - discarding factors / shortening the dual code based on linear OA(81¹⁰, 81, F₈₁, 10) (dual of [81, 71, 11]-code or 81-arc in PG(9,81)), using
        Reedâ€“Solomon code RS(71,81) [i]

(40, 40+31, large)-Net in Base 81 — Upper bound on s

There is no (40, 71, large)-net in base 81, because

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