MinT - Best Known (137−15, 137, s)-Nets in Base 8

Best Known (137−15, 137, s)-Nets in Base 8

(137−15, 137, 2397148)-Net over F₈ — Constructive and digital

Digital (122, 137, 2397148)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (16, 23, 406)-net over F₈, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 1, 73)-net over F₈, using
      - s-reduction based on digital (0, 1, s)-net over F₈ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    2. digital (1, 3, 73)-net over F₈, using
      - kÂ =Â 2 construction [i]
    3. 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]
    4. digital (7, 14, 130)-net over F₈, using
      - trace code for nets [i] based on digital (0, 7, 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 (99, 114, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵⁷, 1198371, F₆₄, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(64⁵⁷, 8388598, F₆₄, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(137−15, 137, 2397265)-Net in Base 8 — Constructive

(122, 137, 2397265)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (16, 23, 523)-net in base 8, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 9)-net over F₈, using
      - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
        the rational function field F₈(x) [i]
        Niederreiter sequence [i]
    2. (13, 20, 514)-net in base 8, using
      - base change [i] based on digital (8, 15, 514)-net over F₁₆, using
        1 times m-reduction [i] based on digital (8, 16, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 8, 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 (99, 114, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵⁷, 1198371, F₆₄, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(64⁵⁷, 8388598, F₆₄, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(137−15, 137, large)-Net over F₈ — Digital

Digital (122, 137, large)-net over F₈, using

t-expansion [i] based on digital (119, 137, large)-net over F₈, using
- 2 times m-reduction [i] based on digital (119, 139, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³⁹, large, F₈, 20) (dual of [large, large−139, 21]-code), using
    - 2 times code embedding in larger space [i] based on linear OA(8¹³⁷, large, F₈, 20) (dual of [large, large−137, 21]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(137−15, 137, large)-Net in Base 8 — Upper bound on s

There is no (122, 137, large)-net in base 8, because

13 times m-reduction [i] would yield (122, 124, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (137−15, 137, s)-Nets in Base 8

(137−15, 137, 2397148)-Net over F8 — Constructive and digital

(137−15, 137, 2397265)-Net in Base 8 — Constructive

(137−15, 137, large)-Net over F8 — Digital

(137−15, 137, large)-Net in Base 8 — Upper bound on s

(137−15, 137, 2397148)-Net over F₈ — Constructive and digital

(137−15, 137, large)-Net over F₈ — Digital