MinT - Best Known (89−16, 89, s)-Nets in Base 16

Best Known (89−16, 89, s)-Nets in Base 16

(89−16, 89, 131136)-Net over F₁₆ — Constructive and digital

Digital (73, 89, 131136)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 14, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]
2. digital (59, 75, 131071)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁷⁵, 131071, F₁₆, 16, 16) (dual of [(131071, 16), 2097061, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(16⁷⁵, 1048568, F₁₆, 16) (dual of [1048568, 1048493, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁷⁵, 1048575, F₁₆, 16) (dual of [1048575, 1048500, 17]-code), using
        1 times truncation [i] based on linear OA(16⁷⁶, 1048576, F₁₆, 17) (dual of [1048576, 1048500, 18]-code), using
        an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(89−16, 89, 262161)-Net in Base 16 — Constructive

(73, 89, 262161)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. (65, 81, 262144)-net in base 16, using
  - net defined by OOA [i] based on OOA(16⁸¹, 262144, S₁₆, 16, 16), using
    - OA 8-folding and stacking [i] based on OA(16⁸¹, 2097152, S₁₆, 16), using
      - discarding factors based on OA(16⁸¹, 2097155, S₁₆, 16), using
        discarding parts of the base [i] based on linear OA(128⁴⁶, 2097155, F₁₂₈, 16) (dual of [2097155, 2097109, 17]-code), using
        constructionÂ X applied to C_e(15) âŠ‚ C_e(14) [i] based on
        linear OA(128⁴⁶, 2097152, F₁₂₈, 16) (dual of [2097152, 2097106, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(128⁴³, 2097152, F₁₂₈, 15) (dual of [2097152, 2097109, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(89−16, 89, 5972019)-Net over F₁₆ — Digital

Digital (73, 89, 5972019)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−16, 89, large)-Net in Base 16 — Upper bound on s

There is no (73, 89, large)-net in base 16, because

14 times m-reduction [i] would yield (73, 75, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (89−16, 89, s)-Nets in Base 16

(89−16, 89, 131136)-Net over F16 — Constructive and digital

(89−16, 89, 262161)-Net in Base 16 — Constructive

(89−16, 89, 5972019)-Net over F16 — Digital

(89−16, 89, large)-Net in Base 16 — Upper bound on s

(89−16, 89, 131136)-Net over F₁₆ — Constructive and digital

(89−16, 89, 5972019)-Net over F₁₆ — Digital