MinT - Best Known (24, 24+7, s)-Nets in Base 81

Best Known (24, 24+7, s)-Nets in Base 81

(24, 24+7, 3334448)-Net over F₈₁ — Constructive and digital

Digital (24, 31, 3334448)-net over F₈₁, using

net defined by OOA [i] based on linear OOA(81³¹, 3334448, F₈₁, 9, 7) (dual of [(3334448, 9), 30010001, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(81³¹, 3334449, F₈₁, 3, 7) (dual of [(3334449, 3), 10003316, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. 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. 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]

(24, 24+7, large)-Net over F₈₁ — Digital

Digital (24, 31, large)-net over F₈₁, using

2 times m-reduction [i] based on digital (24, 33, large)-net over F₈₁, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81³³, large, F₈₁, 9) (dual of [large, large−33, 10]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(24, 24+7, large)-Net in Base 81 — Upper bound on s

There is no (24, 31, large)-net in base 81, because

5 times m-reduction [i] would yield (24, 26, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]