MinT - Best Known (29−7, 29, s)-Nets in Base 81

Best Known (29−7, 29, s)-Nets in Base 81

(29−7, 29, 2802762)-Net over F₈₁ — Constructive and digital

Digital (22, 29, 2802762)-net over F₈₁, using

net defined by OOA [i] based on linear OOA(81²⁹, 2802762, F₈₁, 9, 7) (dual of [(2802762, 9), 25224829, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(81²⁹, 2802763, F₈₁, 3, 7) (dual of [(2802763, 3), 8408260, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(81⁴, 6562, F₈₁, 3, 3) (dual of [(6562, 3), 19682, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(81⁴, 6562, F₈₁, 2, 3) (dual of [(6562, 2), 13120, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁴, 6562, F₈₁, 3) (dual of [6562, 6558, 4]-code or 6562-cap in PG(3,81)), using
        ovoid in PG(3,Â 81) [i]
    2. linear OOA(81²⁵, 2796201, F₈₁, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(81²⁵, large, F₈₁, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(29−7, 29, large)-Net over F₈₁ — Digital

Digital (22, 29, large)-net over F₈₁, using

t-expansion [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]

(29−7, 29, large)-Net in Base 81 — Upper bound on s

There is no (22, 29, large)-net in base 81, because

5 times m-reduction [i] would yield (22, 24, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]