MinT - Best Known (150−13, 150, s)-Nets in Base 8

Best Known (150−13, 150, s)-Nets in Base 8

(150−13, 150, 5594416)-Net over F₈ — Constructive and digital

Digital (137, 150, 5594416)-net over F₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 11, 2016)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹¹, 2016, F₈, 4, 4) (dual of [(2016, 4), 8053, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8¹¹, 2016, F₈, 3, 4) (dual of [(2016, 3), 6037, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(8¹¹, 4032, F₈, 4) (dual of [4032, 4021, 5]-code), using
        base reduction for projective spaces (embedding PG(5,64) in PG(10,8)) [i] based on linear OA(64⁶, 4032, F₆₄, 4) (dual of [4032, 4026, 5]-code), using
        1 times truncation [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (35, 41, 2796200)-net over F₈, using
  - s-reduction based on digital (35, 41, 2796201)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8⁴¹, 2796201, F₈, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(8⁴¹, large, F₈, 6) (dual of [large, large−41, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. digital (85, 98, 2796200)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁸, 2796200, F₈, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁸, 8388601, F₈, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁹⁸, 8388602, F₈, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on linear OOA(64⁴⁹, 4194301, F₆₄, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁹, 8388602, F₆₄, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code 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]

(150−13, 150, large)-Net over F₈ — Digital

Digital (137, 150, large)-net over F₈, using

t-expansion [i] based on digital (131, 150, large)-net over F₈, using
- 3 times m-reduction [i] based on digital (131, 153, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁵³, large, F₈, 22) (dual of [large, large−153, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(150−13, 150, large)-Net in Base 8 — Upper bound on s

There is no (137, 150, large)-net in base 8, because

11 times m-reduction [i] would yield (137, 139, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (150−13, 150, s)-Nets in Base 8

(150−13, 150, 5594416)-Net over F8 — Constructive and digital

(150−13, 150, large)-Net over F8 — Digital

(150−13, 150, large)-Net in Base 8 — Upper bound on s

(150−13, 150, 5594416)-Net over F₈ — Constructive and digital

(150−13, 150, large)-Net over F₈ — Digital