MinT - Best Known (59, 59+21, s)-Nets in Base 81

Best Known (59, 59+21, s)-Nets in Base 81

(59, 59+21, 54456)-Net over F₈₁ — Constructive and digital

Digital (59, 80, 54456)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (9, 19, 1312)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81¹⁹, 1312, F₈₁, 10, 10) (dual of [(1312, 10), 13101, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(81¹⁹, 6560, F₈₁, 10) (dual of [6560, 6541, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(81¹⁹, 6561, F₈₁, 10) (dual of [6561, 6542, 11]-code), using
        an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
2. digital (40, 61, 53144)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁶¹, 53144, F₈₁, 21, 21) (dual of [(53144, 21), 1115963, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(81⁶¹, 531441, F₈₁, 21) (dual of [531441, 531380, 22]-code), using
      - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(59, 59+21, 4468454)-Net over F₈₁ — Digital

Digital (59, 80, 4468454)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+21, large)-Net in Base 81 — Upper bound on s

There is no (59, 80, large)-net in base 81, because

19 times m-reduction [i] would yield (59, 61, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]