MinT - Best Known (65−29, 65, s)-Nets in Base 81

Best Known (65−29, 65, s)-Nets in Base 81

(65−29, 65, 730)-Net over F₈₁ — Constructive and digital

Digital (36, 65, 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]

(65−29, 65, 4549)-Net over F₈₁ — Digital

Digital (36, 65, 4549)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁶⁵, 4549, F₈₁, 29) (dual of [4549, 4484, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁶⁵, 6587, F₈₁, 29) (dual of [6587, 6522, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(19) [i] based on
    1. linear OA(81⁵⁷, 6561, F₈₁, 29) (dual of [6561, 6504, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [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(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]

(65−29, 65, large)-Net in Base 81 — Upper bound on s

There is no (36, 65, large)-net in base 81, because

27 times m-reduction [i] would yield (36, 38, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]