MinT - Best Known (18−7, 18, s)-Nets in Base 16

Best Known (18−7, 18, s)-Nets in Base 16

(18−7, 18, 771)-Net over F₁₆ — Constructive and digital

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]

(18−7, 18, 848)-Net over F₁₆ — Digital

Digital (11, 18, 848)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁸, 848, F₁₆, 7) (dual of [848, 830, 8]-code), using
- 330 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 12 times 0, 1, 80 times 0, 1, 235 times 0) [i] based on linear OA(16¹⁴, 514, F₁₆, 7) (dual of [514, 500, 8]-code), using
  - trace code [i] based on linear OA(256⁷, 257, F₂₅₆, 7) (dual of [257, 250, 8]-code or 257-arc in PG(6,256)), using
    - extended Reedâ€“Solomon code RS_e(250,256) [i]
    - the expurgated narrow-sense BCH-code C(I) with length 257Â | 256²âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
    - algebraic-geometric code AG(F, Q+123P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using the rational function field F₂₅₆(x) [i]
    - algebraic-geometric code AG(F,83P) with degPÂ =Â 3 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257 (see above)
    - algebraic-geometric code AG(F, Q+49P) with degQ = 4 and degPÂ =Â 5 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257 (see above)

(18−7, 18, 806563)-Net in Base 16 — Upper bound on s

There is no (11, 18, 806564)-net in base 16, because

1 times m-reduction [i] would yield (11, 17, 806564)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 295 148762 670063 195631 > 16¹⁷ [i]