MinT - Best Known (102, 102+12, s)-Nets in Base 4

Best Known (102, 102+12, s)-Nets in Base 4

(102, 102+12, 1398105)-Net over F₄ — Constructive and digital

Digital (102, 114, 1398105)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (96, 108, 1398100)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁰⁸, 1398100, F₄, 12, 12) (dual of [(1398100, 12), 16777092, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(4¹⁰⁸, 8388600, F₄, 12) (dual of [8388600, 8388492, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁰⁸, large, F₄, 12) (dual of [large, large−108, 13]-code), using
        the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(102, 102+12, large)-Net over F₄ — Digital

Digital (102, 114, large)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁴, large, F₄, 12) (dual of [large, large−114, 13]-code), using
- 6 times code embedding in larger space [i] based on linear OA(4¹⁰⁸, large, F₄, 12) (dual of [large, large−108, 13]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(102, 102+12, large)-Net in Base 4 — Upper bound on s

There is no (102, 114, large)-net in base 4, because

10 times m-reduction [i] would yield (102, 104, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]