MinT - Best Known (110−12, 110, s)-Nets in Base 8

Best Known (110−12, 110, s)-Nets in Base 8

(110−12, 110, 2796533)-Net over F₈ — Constructive and digital

Digital (98, 110, 2796533)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 20, 333)-net over F₈, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (1, 3, 73)-net over F₈, using
      - kÂ =Â 2 construction [i]
    2. digital (2, 5, 130)-net over F₈, using
      - s-reduction based on digital (2, 5, 208)-net over F₈, using
        net defined by OOA [i] based on linear OOA(8⁵, 208, F₈, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(8⁵, 208, F₈, 2, 3) (dual of [(208, 2), 411, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁵, 208, F₈, 3) (dual of [208, 203, 4]-code or 208-cap in PG(4,8)), using
        caps completed using a computer search [i]
    3. digital (6, 12, 130)-net over F₈, using
      - trace code for nets [i] based on digital (0, 6, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [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]

(110−12, 110, 2796779)-Net in Base 8 — Constructive

(98, 110, 2796779)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (14, 20, 579)-net in base 8, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 65)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8⁴, 65, F₈, 3, 3) (dual of [(65, 3), 191, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(8⁴, 65, F₈, 2, 3) (dual of [(65, 2), 126, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
        ovoid in PG(3,Â 8) [i]
    2. (10, 16, 514)-net in base 8, using
      - base change [i] based on digital (6, 12, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 6, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [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]

(110−12, 110, large)-Net over F₈ — Digital

Digital (98, 110, large)-net over F₈, using

t-expansion [i] based on digital (96, 110, large)-net over F₈, using
- 2 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]

(110−12, 110, large)-Net in Base 8 — Upper bound on s

There is no (98, 110, large)-net in base 8, because

10 times m-reduction [i] would yield (98, 100, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (110−12, 110, s)-Nets in Base 8

(110−12, 110, 2796533)-Net over F8 — Constructive and digital

(110−12, 110, 2796779)-Net in Base 8 — Constructive

(110−12, 110, large)-Net over F8 — Digital

(110−12, 110, large)-Net in Base 8 — Upper bound on s

(110−12, 110, 2796533)-Net over F₈ — Constructive and digital

(110−12, 110, large)-Net over F₈ — Digital