MinT - Best Known (38−5, 38, s)-Nets in Base 81

Best Known (38−5, 38, s)-Nets in Base 81

(38−5, 38, large)-Net over F₈₁ — Constructive and digital

Digital (33, 38, large)-net over F₈₁, using

t-expansion [i] based on digital (32, 38, large)-net over F₈₁, using
- 1 times m-reduction [i] based on digital (32, 39, large)-net over F₈₁, using
  - (u,Â u+v)-construction [i] based on
    1. digital (4, 7, large)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81⁷, large, F₈₁, 3, 3), using
        appending kth column [i] based on linear OOA(81⁷, large, F₈₁, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁷, large, F₈₁, 3) (dual of [large, large−7, 4]-code), using
        product of projective cap with 81²-ovoid cap in AG(3,Â 81) [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. digital (25, 32, 5592401)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81³², 5592401, F₈₁, 9, 7) (dual of [(5592401, 9), 50331577, 8]-NRT-code), using
        OOA stacking with additional row [i] based on linear OOA(81³², 5592402, F₈₁, 3, 7) (dual of [(5592402, 3), 16777174, 8]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(81⁷, large, F₈₁, 3, 3) (see above)
        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]

(38−5, 38, large)-Net in Base 81 — Upper bound on s

There is no (33, 38, large)-net in base 81, because

3 times m-reduction [i] would yield (33, 35, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (38−5, 38, s)-Nets in Base 81

(38−5, 38, large)-Net over F81 — Constructive and digital

(38−5, 38, large)-Net in Base 81 — Upper bound on s

(38−5, 38, large)-Net over F₈₁ — Constructive and digital