MinT - Best Known (35, 48, s)-Nets in Base 81

Best Known (35, 48, s)-Nets in Base 81

(35, 48, 90760)-Net over F₈₁ — Constructive and digital

Digital (35, 48, 90760)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (5, 11, 2187)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81¹¹, 2187, F₈₁, 6, 6) (dual of [(2187, 6), 13111, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(81¹¹, 6561, F₈₁, 6) (dual of [6561, 6550, 7]-code), using
      - an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
2. digital (24, 37, 88573)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81³⁷, 88573, F₈₁, 13, 13) (dual of [(88573, 13), 1151412, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(81³⁷, 531439, F₈₁, 13) (dual of [531439, 531402, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(81³⁷, 531441, F₈₁, 13) (dual of [531441, 531404, 14]-code), using
        an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(35, 48, 2845853)-Net over F₈₁ — Digital

Digital (35, 48, 2845853)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(35, 48, large)-Net in Base 81 — Upper bound on s

There is no (35, 48, large)-net in base 81, because

11 times m-reduction [i] would yield (35, 37, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]