MinT - Best Known (152−12, 152, s)-Nets in Base 8

Best Known (152−12, 152, s)-Nets in Base 8

(152−12, 152, 5854547)-Net over F₈ — Constructive and digital

Digital (140, 152, 5854547)-net over F₈, using

8¹ times duplication [i] based on digital (139, 151, 5854547)-net over F₈, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (16, 20, 262147)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8²⁰, 262147, F₈, 4, 4) (dual of [(262147, 4), 1048568, 5]-NRT-code), using
      - appending kth column [i] based on linear OOA(8²⁰, 262147, F₈, 3, 4) (dual of [(262147, 3), 786421, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(8²⁰, 524294, F₈, 4) (dual of [524294, 524274, 5]-code), using
        trace code [i] based on linear OA(64¹⁰, 262147, F₆₄, 4) (dual of [262147, 262137, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(64¹⁰, 262144, F₆₄, 4) (dual of [262144, 262134, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(64⁷, 262144, F₆₄, 3) (dual of [262144, 262137, 4]-code or 262144-cap in PG(6,64)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 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]

(152−12, 152, large)-Net over F₈ — Digital

Digital (140, 152, large)-net over F₈, using

t-expansion [i] based on digital (138, 152, large)-net over F₈, using
- 9 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]

(152−12, 152, large)-Net in Base 8 — Upper bound on s

There is no (140, 152, large)-net in base 8, because

10 times m-reduction [i] would yield (140, 142, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (152−12, 152, s)-Nets in Base 8

(152−12, 152, 5854547)-Net over F8 — Constructive and digital

(152−12, 152, large)-Net over F8 — Digital

(152−12, 152, large)-Net in Base 8 — Upper bound on s

(152−12, 152, 5854547)-Net over F₈ — Constructive and digital

(152−12, 152, large)-Net over F₈ — Digital