MinT - Best Known (72−12, 72, s)-Nets in Base 81

Best Known (72−12, 72, s)-Nets in Base 81

(72−12, 72, 2799440)-Net over F₈₁ — Constructive and digital

Digital (60, 72, 2799440)-net over F₈₁, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 6, 3240)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁶, 3240, F₈₁, 4, 4) (dual of [(3240, 4), 12954, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(81⁶, 6480, F₈₁, 4) (dual of [6480, 6474, 5]-code), using
      - 1 times truncation [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (15, 21, 1398100)-net over F₈₁, using
  - s-reduction based on digital (15, 21, 2796201)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81²¹, 2796201, F₈₁, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(81²¹, large, F₈₁, 6) (dual of [large, large−21, 7]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. digital (33, 45, 1398100)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁴⁵, 1398100, F₈₁, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(81⁴⁵, 8388600, F₈₁, 12) (dual of [8388600, 8388555, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁴⁵, large, F₈₁, 12) (dual of [large, large−45, 13]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(72−12, 72, large)-Net over F₈₁ — Digital

Digital (60, 72, large)-net over F₈₁, using

9 times m-reduction [i] based on digital (60, 81, large)-net over F₈₁, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁸¹, large, F₈₁, 21) (dual of [large, large−81, 22]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(72−12, 72, large)-Net in Base 81 — Upper bound on s

There is no (60, 72, large)-net in base 81, because

10 times m-reduction [i] would yield (60, 62, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (72−12, 72, s)-Nets in Base 81

(72−12, 72, 2799440)-Net over F81 — Constructive and digital

(72−12, 72, large)-Net over F81 — Digital

(72−12, 72, large)-Net in Base 81 — Upper bound on s

(72−12, 72, 2799440)-Net over F₈₁ — Constructive and digital

(72−12, 72, large)-Net over F₈₁ — Digital