MinT - Best Known (10, 10+7, s)-Nets in Base 3

Best Known (10, 10+7, s)-Nets in Base 3

(10, 10+7, 40)-Net over F₃ — Constructive and digital

Digital (10, 17, 40)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 5, 20)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁵, 20, F₃, 3, 3) (dual of [(20, 3), 55, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(3⁵, 20, F₃, 2, 3) (dual of [(20, 2), 35, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁵, 20, F₃, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
        doubling the cap [i] based on linear OA(3⁴, 10, F₃, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
        ovoid in PG(3,Â 3) [i]
2. digital (5, 12, 20)-net over F₃, using
  - linear code embedding found in a computer search [i]

(10, 10+7, 44)-Net over F₃ — Digital

Digital (10, 17, 44)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷, 44, F₃, 2, 7) (dual of [(44, 2), 71, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁷, 88, F₃, 7) (dual of [88, 71, 8]-code), using
  - a â€œGraâ€ code from Grasslâ€™s database [i]

(10, 10+7, 315)-Net in Base 3 — Upper bound on s

There is no (10, 17, 316)-net in base 3, because

1 times m-reduction [i] would yield (10, 16, 316)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 43 075857 > 3¹⁶ [i]