MinT - Best Known (21, 27, s)-Nets in Base 81

Best Known (21, 27, s)-Nets in Base 81

(21, 27, 3334449)-Net over F₈₁ — Constructive and digital

Digital (21, 27, 3334449)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (3, 6, 538248)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁶, 538248, F₈₁, 3, 3) (dual of [(538248, 3), 1614738, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(81⁶, 538248, F₈₁, 2, 3) (dual of [(538248, 2), 1076490, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁶, 538248, F₈₁, 3) (dual of [538248, 538242, 4]-code or 538248-cap in PG(5,81)), using
        large caps in PG(5,Â b) [i]
2. 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]

(21, 27, large)-Net over F₈₁ — Digital

Digital (21, 27, large)-net over F₈₁, using

2 times m-reduction [i] based on digital (21, 29, large)-net over F₈₁, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81²⁹, large, F₈₁, 8) (dual of [large, large−29, 9]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(21, 27, large)-Net in Base 81 — Upper bound on s

There is no (21, 27, large)-net in base 81, because

4 times m-reduction [i] would yield (21, 23, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]