MinT - Best Known (80−12, 80, s)-Nets in Base 8

Best Known (80−12, 80, s)-Nets in Base 8

(80−12, 80, 349550)-Net over F₈ — Constructive and digital

Digital (68, 80, 349550)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 9, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (59, 71, 349526)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁷¹, 349526, F₈, 12, 12) (dual of [(349526, 12), 4194241, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(8⁷¹, 2097156, F₈, 12) (dual of [2097156, 2097085, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁷¹, 2097159, F₈, 12) (dual of [2097159, 2097088, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(8⁷¹, 2097152, F₈, 12) (dual of [2097152, 2097081, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(8⁶⁴, 2097152, F₈, 11) (dual of [2097152, 2097088, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 8⁷âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [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]

(80−12, 80, 699051)-Net in Base 8 — Constructive

(68, 80, 699051)-net in base 8, using

net defined by OOA [i] based on OOA(8⁸⁰, 699051, S₈, 12, 12), using
- OA 6-folding and stacking [i] based on OA(8⁸⁰, 4194306, S₈, 12), using
  - discarding factors based on OA(8⁸⁰, 4194310, S₈, 12), using
    - trace code [i] based on OA(64⁴⁰, 2097155, S₆₄, 12), using
      - discarding parts of the base [i] based on linear OA(128³⁴, 2097155, F₁₂₈, 12) (dual of [2097155, 2097121, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(128³⁴, 2097152, F₁₂₈, 12) (dual of [2097152, 2097118, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(128³¹, 2097152, F₁₂₈, 11) (dual of [2097152, 2097121, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(80−12, 80, 2593249)-Net over F₈ — Digital

Digital (68, 80, 2593249)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(80−12, 80, large)-Net in Base 8 — Upper bound on s

There is no (68, 80, large)-net in base 8, because

10 times m-reduction [i] would yield (68, 70, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (80−12, 80, s)-Nets in Base 8

(80−12, 80, 349550)-Net over F8 — Constructive and digital

(80−12, 80, 699051)-Net in Base 8 — Constructive

(80−12, 80, 2593249)-Net over F8 — Digital

(80−12, 80, large)-Net in Base 8 — Upper bound on s

(80−12, 80, 349550)-Net over F₈ — Constructive and digital

(80−12, 80, 2593249)-Net over F₈ — Digital