MinT - Best Known (150−12, 150, s)-Nets in Base 8

Best Known (150−12, 150, s)-Nets in Base 8

(150−12, 150, 5723475)-Net over F₈ — Constructive and digital

Digital (138, 150, 5723475)-net over F₈, using

generalized (u,Â u+v)-construction [i] based on
1. digital (15, 19, 131075)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁹, 131075, F₈, 4, 4) (dual of [(131075, 4), 524281, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8¹⁹, 131075, F₈, 3, 4) (dual of [(131075, 3), 393206, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(8¹⁹, 262150, F₈, 4) (dual of [262150, 262131, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(8¹⁹, 262144, F₈, 4) (dual of [262144, 262125, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(8¹³, 262144, F₈, 3) (dual of [262144, 262131, 4]-code or 262144-cap in PG(12,8)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(8⁰, 6, F₈, 0) (dual of [6, 6, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [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 (78, 90, 2796200)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁰, 2796200, F₈, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁰, 8388601, F₈, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁹⁰, 8388602, F₈, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
        trace code [i] based on linear OOA(64⁴⁵, 4194301, F₆₄, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁵, 8388602, F₆₄, 12) (dual of [8388602, 8388557, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(150−12, 150, large)-Net over F₈ — Digital

Digital (138, 150, large)-net over F₈, using

11 times m-reduction [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]

(150−12, 150, large)-Net in Base 8 — Upper bound on s

There is no (138, 150, large)-net in base 8, because

10 times m-reduction [i] would yield (138, 140, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (150−12, 150, s)-Nets in Base 8

(150−12, 150, 5723475)-Net over F8 — Constructive and digital

(150−12, 150, large)-Net over F8 — Digital

(150−12, 150, large)-Net in Base 8 — Upper bound on s

(150−12, 150, 5723475)-Net over F₈ — Constructive and digital

(150−12, 150, large)-Net over F₈ — Digital