MinT - Best Known (38−19, 38, s)-Nets in Base 81

Best Known (38−19, 38, s)-Nets in Base 81

(38−19, 38, 729)-Net over F₈₁ — Constructive and digital

Digital (19, 38, 729)-net over F₈₁, using

81¹ times duplication [i] based on digital (18, 37, 729)-net over F₈₁, using
- net defined by OOA [i] based on linear OOA(81³⁷, 729, F₈₁, 19, 19) (dual of [(729, 19), 13814, 20]-NRT-code), using
  - OOA 9-folding and stacking with additional row [i] based on linear OA(81³⁷, 6562, F₈₁, 19) (dual of [6562, 6525, 20]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(38−19, 38, 2189)-Net over F₈₁ — Digital

Digital (19, 38, 2189)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(81³⁸, 2189, F₈₁, 3, 19) (dual of [(2189, 3), 6529, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(81³⁸, 6567, F₈₁, 19) (dual of [6567, 6529, 20]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,8]) [i] based on
    1. linear OA(81³⁷, 6562, F₈₁, 19) (dual of [6562, 6525, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(81³³, 6562, F₈₁, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (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]

(38−19, 38, 3636260)-Net in Base 81 — Upper bound on s

There is no (19, 38, 3636261)-net in base 81, because

1 times m-reduction [i] would yield (19, 37, 3636261)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 41109 926747 576562 281652 254435 061125 219554 717230 580321 508915 735860 962321 > 81³⁷ [i]