MinT - Best Known (103−13, 103, s)-Nets in Base 8

Best Known (103−13, 103, s)-Nets in Base 8

(103−13, 103, 2796201)-Net over F₈ — Constructive and digital

Digital (90, 103, 2796201)-net over F₈, using

8² times duplication [i] based on digital (88, 101, 2796201)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹⁰¹, 2796201, F₈, 15, 13) (dual of [(2796201, 15), 41942914, 14]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OOA(8¹⁰¹, 5592403, F₈, 3, 13) (dual of [(5592403, 3), 16777108, 14]-NRT-code), using
    - 1 times NRT-code embedding in larger space [i] based on linear OOA(8⁹⁸, 5592402, F₈, 3, 13) (dual of [(5592402, 3), 16777108, 14]-NRT-code), using
      - trace code [i] based on linear OOA(64⁴⁹, 2796201, F₆₄, 3, 13) (dual of [(2796201, 3), 8388554, 14]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(103−13, 103, large)-Net over F₈ — Digital

Digital (90, 103, large)-net over F₈, using

2 times m-reduction [i] based on digital (90, 105, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁵, large, F₈, 15) (dual of [large, large−105, 16]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(103−13, 103, large)-Net in Base 8 — Upper bound on s

There is no (90, 103, large)-net in base 8, because

11 times m-reduction [i] would yield (90, 92, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]