MinT - Best Known (38, 38+37, s)-Nets in Base 81

Best Known (38, 38+37, s)-Nets in Base 81

(38, 38+37, 730)-Net over F₈₁ — Constructive and digital

Digital (38, 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]

(38, 38+37, 2189)-Net over F₈₁ — Digital

Digital (38, 75, 2189)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(81⁷⁵, 2189, F₈₁, 3, 37) (dual of [(2189, 3), 6492, 38]-NRT-code), using
- 81¹ times duplication [i] based on linear OOA(81⁷⁴, 2189, F₈₁, 3, 37) (dual of [(2189, 3), 6493, 38]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(81⁷⁴, 6567, F₈₁, 37) (dual of [6567, 6493, 38]-code), using
    - constructionÂ X applied to C([0,18]) âŠ‚ C([0,17]) [i] based on
      1. linear OA(81⁷³, 6562, F₈₁, 37) (dual of [6562, 6489, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
      2. linear OA(81⁶⁹, 6562, F₈₁, 35) (dual of [6562, 6493, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
      3. linear OA(81¹, 5, F₈₁, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(38, 38+37, 6622675)-Net in Base 81 — Upper bound on s

There is no (38, 75, 6622676)-net in base 81, because

1 times m-reduction [i] would yield (38, 74, 6622676)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 1690 021385 578635 117908 591589 074264 923661 012184 343540 414265 728064 163197 218349 050094 087311 958226 601656 076546 488326 217997 239538 926804 661176 064641 > 81⁷⁴ [i]