MinT - Best Known (54−13, 54, s)-Nets in Base 8

Best Known (54−13, 54, s)-Nets in Base 8

(54−13, 54, 1367)-Net over F₈ — Constructive and digital

Digital (41, 54, 1367)-net over F₈, using

8² times duplication [i] based on digital (39, 52, 1367)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁵², 1367, F₈, 13, 13) (dual of [(1367, 13), 17719, 14]-NRT-code), using
  - OOA 6-folding and stacking with additional row [i] based on linear OA(8⁵², 8203, F₈, 13) (dual of [8203, 8151, 14]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁵², 8204, F₈, 13) (dual of [8204, 8152, 14]-code), using
      - trace code [i] based on linear OA(64²⁶, 4102, F₆₄, 13) (dual of [4102, 4076, 14]-code), using
        constructionÂ X applied to C([0,6]) âŠ‚ C([0,5]) [i] based on
        linear OA(64²⁵, 4097, F₆₄, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(64²¹, 4097, F₆₄, 11) (dual of [4097, 4076, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
        linear OA(64¹, 5, F₆₄, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(54−13, 54, 8759)-Net over F₈ — Digital

Digital (41, 54, 8759)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−13, 54, large)-Net in Base 8 — Upper bound on s

There is no (41, 54, large)-net in base 8, because

11 times m-reduction [i] would yield (41, 43, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]