MinT - Best Known (108−12, 108, s)-Nets in Base 8

Best Known (108−12, 108, s)-Nets in Base 8

(108−12, 108, 2796460)-Net over F₈ — Constructive and digital

Digital (96, 108, 2796460)-net over F₈, using

8¹ times duplication [i] based on digital (95, 107, 2796460)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 17, 260)-net over F₈, using
    - (u,Â u+v)-construction [i] based on
      1. 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]
      2. 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]

(108−12, 108, 2796714)-Net in Base 8 — Constructive

(96, 108, 2796714)-net in base 8, using

base change [i] based on digital (69, 81, 2796714)-net over F₁₆, using
- 16¹ times duplication [i] based on digital (68, 80, 2796714)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. 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 (56, 68, 2796200)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16⁶⁸, 2796200, F₁₆, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁶⁸, 8388601, F₁₆, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁶⁸, 8388602, F₁₆, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
        trace code [i] based on linear OOA(256³⁴, 4194301, F₂₅₆, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁴, 8388602, F₂₅₆, 12) (dual of [8388602, 8388568, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(108−12, 108, large)-Net over F₈ — Digital

Digital (96, 108, large)-net over F₈, using

4 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]

(108−12, 108, large)-Net in Base 8 — Upper bound on s

There is no (96, 108, large)-net in base 8, because

10 times m-reduction [i] would yield (96, 98, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (108−12, 108, s)-Nets in Base 8

(108−12, 108, 2796460)-Net over F8 — Constructive and digital

(108−12, 108, 2796714)-Net in Base 8 — Constructive

(108−12, 108, large)-Net over F8 — Digital

(108−12, 108, large)-Net in Base 8 — Upper bound on s

(108−12, 108, 2796460)-Net over F₈ — Constructive and digital

(108−12, 108, large)-Net over F₈ — Digital