MinT - Best Known (99, 99+12, s)-Nets in Base 8

Best Known (99, 99+12, s)-Nets in Base 8

(99, 99+12, 2797566)-Net over F₈ — Constructive and digital

Digital (99, 111, 2797566)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (15, 21, 1366)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²¹, 1366, F₈, 6, 6) (dual of [(1366, 6), 8175, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(8²¹, 4098, F₈, 6) (dual of [4098, 4077, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(8²¹, 4100, F₈, 6) (dual of [4100, 4079, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(8²¹, 4096, F₈, 6) (dual of [4096, 4075, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(8¹⁷, 4096, F₈, 5) (dual of [4096, 4079, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 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]

(99, 99+12, large)-Net over F₈ — Digital

Digital (99, 111, large)-net over F₈, using

t-expansion [i] based on digital (96, 111, large)-net over F₈, using
- 1 times m-reduction [i] based on digital (96, 112, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹², large, F₈, 16) (dual of [large, large−112, 17]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(99, 99+12, large)-Net in Base 8 — Upper bound on s

There is no (99, 111, large)-net in base 8, because

10 times m-reduction [i] would yield (99, 101, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (99, 99+12, s)-Nets in Base 8

(99, 99+12, 2797566)-Net over F8 — Constructive and digital

(99, 99+12, large)-Net over F8 — Digital

(99, 99+12, large)-Net in Base 8 — Upper bound on s

(99, 99+12, 2797566)-Net over F₈ — Constructive and digital

(99, 99+12, large)-Net over F₈ — Digital