MinT - Best Known (72, 72+8, s)-Nets in Base 8

Best Known (72, 72+8, s)-Nets in Base 8

(72, 72+8, 5242879)-Net over F₈ — Constructive and digital

Digital (72, 80, 5242879)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (18, 22, 1048579)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²², 1048579, F₈, 4, 4) (dual of [(1048579, 4), 4194294, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8²², 1048579, F₈, 3, 4) (dual of [(1048579, 3), 3145715, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(8²², 2097158, F₈, 4) (dual of [2097158, 2097136, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(8²², 2097159, F₈, 4) (dual of [2097159, 2097137, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(8²², 2097152, F₈, 4) (dual of [2097152, 2097130, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(8¹⁵, 2097152, F₈, 3) (dual of [2097152, 2097137, 4]-code or 2097152-cap in PG(14,8)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(8⁰, 7, F₈, 0) (dual of [7, 7, 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 (50, 58, 4194300)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁵⁸, 4194300, F₈, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁵⁸, 8388601, F₈, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁵⁸, 8388602, F₈, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
        trace code [i] based on linear OOA(64²⁹, 4194301, F₆₄, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64²⁹, 8388602, F₆₄, 8) (dual of [8388602, 8388573, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(64²⁹, large, F₆₄, 8) (dual of [large, large−29, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(72, 72+8, large)-Net over F₈ — Digital

Digital (72, 80, large)-net over F₈, using

t-expansion [i] based on digital (69, 80, large)-net over F₈, using
- 1 times m-reduction [i] based on digital (69, 81, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸¹, large, F₈, 12) (dual of [large, large−81, 13]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(72, 72+8, large)-Net in Base 8 — Upper bound on s

There is no (72, 80, large)-net in base 8, because

6 times m-reduction [i] would yield (72, 74, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (72, 72+8, s)-Nets in Base 8

(72, 72+8, 5242879)-Net over F8 — Constructive and digital

(72, 72+8, large)-Net over F8 — Digital

(72, 72+8, large)-Net in Base 8 — Upper bound on s

(72, 72+8, 5242879)-Net over F₈ — Constructive and digital

(72, 72+8, large)-Net over F₈ — Digital