MinT - Best Known (75−33, 75, s)-Nets in Base 81

Best Known (75−33, 75, s)-Nets in Base 81

(75−33, 75, 730)-Net over F₈₁ — Constructive and digital

Digital (42, 75, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 75, 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]

(75−33, 75, 5566)-Net over F₈₁ — Digital

Digital (42, 75, 5566)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷⁵, 5566, F₈₁, 33) (dual of [5566, 5491, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁷⁵, 6593, F₈₁, 33) (dual of [6593, 6518, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(21) [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(81⁴³, 6561, F₈₁, 22) (dual of [6561, 6518, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [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]

(75−33, 75, large)-Net in Base 81 — Upper bound on s

There is no (42, 75, large)-net in base 81, because

31 times m-reduction [i] would yield (42, 44, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]