MinT - Best Known (88−14, 88, s)-Nets in Base 16

Best Known (88−14, 88, s)-Nets in Base 16

(88−14, 88, 2396766)-Net over F₁₆ — Constructive and digital

Digital (74, 88, 2396766)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (66, 80, 2396742)-net over F₁₆, using
  - trace code for nets [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(88−14, 88, large)-Net over F₁₆ — Digital

Digital (74, 88, large)-net over F₁₆, using

2 times m-reduction [i] based on digital (74, 90, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁰, large, F₁₆, 16) (dual of [large, large−90, 17]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(88−14, 88, large)-Net in Base 16 — Upper bound on s

There is no (74, 88, large)-net in base 16, because

12 times m-reduction [i] would yield (74, 76, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]