MinT - Best Known (43, 79, s)-Nets in Base 81

Best Known (43, 79, s)-Nets in Base 81

(43, 79, 730)-Net over F₈₁ — Constructive and digital

Digital (43, 79, 730)-net over F₈₁, using

t-expansion [i] based on digital (36, 79, 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]

(43, 79, 4027)-Net over F₈₁ — Digital

Digital (43, 79, 4027)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁷⁹, 4027, F₈₁, 36) (dual of [4027, 3948, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(81⁷⁹, 6587, F₈₁, 36) (dual of [6587, 6508, 37]-code), using
  - constructionÂ X applied to C_e(35) âŠ‚ C_e(26) [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(81⁵³, 6561, F₈₁, 27) (dual of [6561, 6508, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [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]

(43, 79, large)-Net in Base 81 — Upper bound on s

There is no (43, 79, large)-net in base 81, because

34 times m-reduction [i] would yield (43, 45, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]