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

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

(81−36, 81, 730)-Net over F₈₁ — Constructive and digital

Digital (45, 81, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 81, 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]

(81−36, 81, 5219)-Net over F₈₁ — Digital

Digital (45, 81, 5219)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁸¹, 5219, F₈₁, 36) (dual of [5219, 5138, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁸¹, 6593, F₈₁, 36) (dual of [6593, 6512, 37]-code), using
  - constructionÂ X applied to C_e(35) âŠ‚ C_e(24) [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(81⁴⁹, 6561, F₈₁, 25) (dual of [6561, 6512, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [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]

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

There is no (45, 81, large)-net in base 81, because

34 times m-reduction [i] would yield (45, 47, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]