MinT - Best Known (116−12, 116, s)-Nets in Base 8

Best Known (116−12, 116, s)-Nets in Base 8

(116−12, 116, 2807124)-Net over F₈ — Constructive and digital

Digital (104, 116, 2807124)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (20, 26, 10924)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²⁶, 10924, F₈, 6, 6) (dual of [(10924, 6), 65518, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(8²⁶, 32772, F₈, 6) (dual of [32772, 32746, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(8²⁶, 32773, F₈, 6) (dual of [32773, 32747, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(8²⁶, 32768, F₈, 6) (dual of [32768, 32742, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(8²¹, 32768, F₈, 5) (dual of [32768, 32747, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(8⁰, 5, F₈, 0) (dual of [5, 5, 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 (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]

(116−12, 116, large)-Net over F₈ — Digital

Digital (104, 116, large)-net over F₈, using

t-expansion [i] based on digital (100, 116, large)-net over F₈, using
- 1 times m-reduction [i] based on digital (100, 117, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁷, large, F₈, 17) (dual of [large, large−117, 18]-code), using
    - 4 times code embedding in larger space [i] based on linear OA(8¹¹³, large, F₈, 17) (dual of [large, large−113, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(116−12, 116, large)-Net in Base 8 — Upper bound on s

There is no (104, 116, large)-net in base 8, because

10 times m-reduction [i] would yield (104, 106, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (116−12, 116, s)-Nets in Base 8

(116−12, 116, 2807124)-Net over F8 — Constructive and digital

(116−12, 116, large)-Net over F8 — Digital

(116−12, 116, large)-Net in Base 8 — Upper bound on s

(116−12, 116, 2807124)-Net over F₈ — Constructive and digital

(116−12, 116, large)-Net over F₈ — Digital