MinT - Best Known (31, 31+4, s)-Nets in Base 8

Best Known (31, 31+4, s)-Nets in Base 8

(31, 31+4, large)-Net over F₈ — Constructive and digital

Digital (31, 35, large)-net over F₈, using

1 times m-reduction [i] based on digital (31, 36, large)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 2, 9)-net over F₈, using
    - kÂ =Â 2 construction [i]
  2. digital (29, 34, 8388602)-net over F₈, using
    - trace code for nets [i] based on digital (12, 17, 4194301)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64¹⁷, 4194301, F₆₄, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(64¹⁷, large, F₆₄, 5) (dual of [large, large−17, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]

(31, 31+4, large)-Net in Base 8 — Upper bound on s

There is no (31, 35, large)-net in base 8, because

2 times m-reduction [i] would yield (31, 33, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]