MinT - Best Known (116, 130, s)-Nets in Base 8

Best Known (116, 130, s)-Nets in Base 8

(116, 130, 2397262)-Net over F₈ — Constructive and digital

Digital (116, 130, 2397262)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (17, 24, 520)-net over F₈, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 65)-net over F₈, using
      - s-reduction based on digital (0, 0, s)-net over F₈ with arbitrarily large s, using
        arbitrary set of 8⁰ points [i]
    2. digital (0, 1, 65)-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]
    3. digital (0, 1, 65)-net over F₈ (see above)
    4. digital (0, 1, 65)-net over F₈ (see above)
    5. digital (0, 1, 65)-net over F₈ (see above)
    6. digital (1, 3, 65)-net over F₈, using
      - s-reduction based on digital (1, 3, 73)-net over F₈, using
        kÂ =Â 2 construction [i]
    7. 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]
    8. digital (6, 13, 65)-net over F₈, using
      - base reduction for projective spaces (embedding PG(6,64) in PG(12,8)) 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 (92, 106, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(116, 130, 2397513)-Net in Base 8 — Constructive

(116, 130, 2397513)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (17, 24, 771)-net in base 8, using
  - base change [i] based on digital (11, 18, 771)-net over F₁₆, using
    - (u,Â u+v)-construction [i] based on
      1. digital (1, 4, 257)-net over F₁₆, using
        net defined by OOA [i] based on linear OOA(16⁴, 257, F₁₆, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(16⁴, 257, F₁₆, 2, 3) (dual of [(257, 2), 510, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(16⁴, 257, F₁₆, 3) (dual of [257, 253, 4]-code or 257-cap in PG(3,16)), using
        ovoid in PG(3,Â 16) [i]
      2. digital (7, 14, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 7, 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 (92, 106, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(116, 130, large)-Net over F₈ — Digital

Digital (116, 130, large)-net over F₈, using

t-expansion [i] based on digital (113, 130, large)-net over F₈, using
- 2 times m-reduction [i] based on digital (113, 132, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹³², large, F₈, 19) (dual of [large, large−132, 20]-code), using
    - 3 times code embedding in larger space [i] based on linear OA(8¹²⁹, large, F₈, 19) (dual of [large, large−129, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(116, 130, large)-Net in Base 8 — Upper bound on s

There is no (116, 130, large)-net in base 8, because

12 times m-reduction [i] would yield (116, 118, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (116, 130, s)-Nets in Base 8

(116, 130, 2397262)-Net over F8 — Constructive and digital

(116, 130, 2397513)-Net in Base 8 — Constructive

(116, 130, large)-Net over F8 — Digital

(116, 130, large)-Net in Base 8 — Upper bound on s

(116, 130, 2397262)-Net over F₈ — Constructive and digital

(116, 130, large)-Net over F₈ — Digital