MinT - Best Known (82−36, 82, s)-Nets in Base 81

Best Known (82−36, 82, s)-Nets in Base 81

(82−36, 82, 730)-Net over F₈₁ — Constructive and digital

Digital (46, 82, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 82, 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]

(82−36, 82, 5941)-Net over F₈₁ — Digital

Digital (46, 82, 5941)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁸², 5941, F₈₁, 36) (dual of [5941, 5859, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁸², 6596, F₈₁, 36) (dual of [6596, 6514, 37]-code), using
  - constructionÂ X applied to C_e(35) âŠ‚ C_e(23) [i] based on
    1. linear OA(81⁷¹, 6561, F₈₁, 36) (dual of [6561, 6490, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [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(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]

(82−36, 82, large)-Net in Base 81 — Upper bound on s

There is no (46, 82, large)-net in base 81, because

34 times m-reduction [i] would yield (46, 48, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]