MinT - Best Known (162−19, 162, s)-Nets in Base 8

Best Known (162−19, 162, s)-Nets in Base 8

(162−19, 162, 1864169)-Net over F₈ — Constructive and digital

Digital (143, 162, 1864169)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 16, 36)-net over F₈, using
  - 1 times m-reduction [i] based on digital (7, 17, 36)-net over F₈, using
    - Niederreiter sequence (Bratleyâ€“Foxâ€“Niederreiter implementation) [i]
2. digital (127, 146, 1864133)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴⁶, 1864133, F₈, 21, 19) (dual of [(1864133, 21), 39146647, 20]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8¹⁴⁶, 5592400, F₈, 3, 19) (dual of [(5592400, 3), 16777054, 20]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8¹⁴⁶, 5592402, F₈, 3, 19) (dual of [(5592402, 3), 16777060, 20]-NRT-code), using
        trace code [i] based on linear OOA(64⁷³, 2796201, F₆₄, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64⁷³, large, F₆₄, 19) (dual of [large, large−73, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(162−19, 162, 1864171)-Net in Base 8 — Constructive

(143, 162, 1864171)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (7, 16, 38)-net in base 8, using
  - base change [i] based on digital (3, 12, 38)-net over F₁₆, using
    - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (127, 146, 1864133)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴⁶, 1864133, F₈, 21, 19) (dual of [(1864133, 21), 39146647, 20]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8¹⁴⁶, 5592400, F₈, 3, 19) (dual of [(5592400, 3), 16777054, 20]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8¹⁴⁶, 5592402, F₈, 3, 19) (dual of [(5592402, 3), 16777060, 20]-NRT-code), using
        trace code [i] based on linear OOA(64⁷³, 2796201, F₆₄, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64⁷³, large, F₆₄, 19) (dual of [large, large−73, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(162−19, 162, large)-Net over F₈ — Digital

Digital (143, 162, large)-net over F₈, using

8¹ times duplication [i] based on digital (142, 161, large)-net over F₈, using
- t-expansion [i] based on digital (138, 161, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶¹, large, F₈, 23) (dual of [large, large−161, 24]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(162−19, 162, large)-Net in Base 8 — Upper bound on s

There is no (143, 162, large)-net in base 8, because

17 times m-reduction [i] would yield (143, 145, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]