MinT - Best Known (24−9, 24, s)-Nets in Base 16

Best Known (24−9, 24, s)-Nets in Base 16

(24−9, 24, 634)-Net over F₁₆ — Constructive and digital

Digital (15, 24, 634)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 120)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁶, 120, F₁₆, 4, 4) (dual of [(120, 4), 474, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(16⁶, 240, F₁₆, 4) (dual of [240, 234, 5]-code), using
      - 1 times truncation [i] based on linear OA(16⁷, 241, F₁₆, 5) (dual of [241, 234, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (9, 18, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 9, 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]

(24−9, 24, 1036)-Net over F₁₆ — Digital

Digital (15, 24, 1036)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16²⁴, 1036, F₁₆, 9) (dual of [1036, 1012, 10]-code), using
- 516 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 16 times 0, 1, 58 times 0, 1, 154 times 0, 1, 280 times 0) [i] based on linear OA(16¹⁸, 514, F₁₆, 9) (dual of [514, 496, 10]-code), using
  - trace code [i] based on linear OA(256⁹, 257, F₂₅₆, 9) (dual of [257, 248, 10]-code or 257-arc in PG(8,256)), using
    - extended Reedâ€“Solomon code RS_e(248,256) [i]
    - the expurgated narrow-sense BCH-code C(I) with length 257Â | 256²âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
    - algebraic-geometric code AG(F, Q+122P) 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, Q+81P) with degQ = 4 and 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 = 2 and degPÂ =Â 5 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257 (see above)

(24−9, 24, 1237801)-Net in Base 16 — Upper bound on s

There is no (15, 24, 1237802)-net in base 16, because

1 times m-reduction [i] would yield (15, 23, 1237802)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 4951 776041 460903 940097 673046 > 16²³ [i]