MinT - Best Known (115−13, 115, s)-Nets in Base 8

Best Known (115−13, 115, s)-Nets in Base 8

(115−13, 115, 2796460)-Net over F₈ — Constructive and digital

Digital (102, 115, 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 (85, 98, 2796200)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁹⁸, 2796200, F₈, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁸, 8388601, F₈, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁹⁸, 8388602, F₈, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on linear OOA(64⁴⁹, 4194301, F₆₄, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁹, 8388602, F₆₄, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(115−13, 115, 2796714)-Net in Base 8 — Constructive

(102, 115, 2796714)-net in base 8, using

8¹ times duplication [i] based on (101, 114, 2796714)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (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 (85, 98, 2796200)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8⁹⁸, 2796200, F₈, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁸, 8388601, F₈, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(8⁹⁸, 8388602, F₈, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on linear OOA(64⁴⁹, 4194301, F₆₄, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁹, 8388602, F₆₄, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(115−13, 115, large)-Net over F₈ — Digital

Digital (102, 115, large)-net over F₈, using

t-expansion [i] based on digital (100, 115, large)-net over F₈, using
- 2 times m-reduction [i] based on digital (100, 117, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁷, large, F₈, 17) (dual of [large, large−117, 18]-code), using
    - 4 times code embedding in larger space [i] based on linear OA(8¹¹³, large, F₈, 17) (dual of [large, large−113, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(115−13, 115, large)-Net in Base 8 — Upper bound on s

There is no (102, 115, large)-net in base 8, because

11 times m-reduction [i] would yield (102, 104, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (115−13, 115, s)-Nets in Base 8

(115−13, 115, 2796460)-Net over F8 — Constructive and digital

(115−13, 115, 2796714)-Net in Base 8 — Constructive

(115−13, 115, large)-Net over F8 — Digital

(115−13, 115, large)-Net in Base 8 — Upper bound on s

(115−13, 115, 2796460)-Net over F₈ — Constructive and digital

(115−13, 115, large)-Net over F₈ — Digital