MinT - Best Known (150−12, 150, s)-Nets in Base 4

Best Known (150−12, 150, s)-Nets in Base 4

(150−12, 150, 5592448)-Net over F₄ — Constructive and digital

Digital (138, 150, 5592448)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (8, 14, 48)-net over F₄, using
  - trace code for nets [i] based on digital (1, 7, 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 (124, 136, 5592400)-net over F₄, using
  - trace code for nets [i] based on digital (56, 68, 2796200)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁶⁸, 2796200, F₁₆, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁶⁸, 8388601, F₁₆, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁶⁸, 8388602, F₁₆, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
        trace code [i] based on linear OOA(256³⁴, 4194301, F₂₅₆, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁴, 8388602, F₂₅₆, 12) (dual of [8388602, 8388568, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(150−12, 150, large)-Net over F₄ — Digital

Digital (138, 150, large)-net over F₄, using

4⁵ times duplication [i] based on digital (133, 145, large)-net over F₄, using
- t-expansion [i] based on digital (130, 145, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴⁵, large, F₄, 15) (dual of [large, large−145, 16]-code), using
    - strength reduction [i] based on linear OA(4¹⁴⁵, large, F₄, 17) (dual of [large, large−145, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(150−12, 150, large)-Net in Base 4 — Upper bound on s

There is no (138, 150, large)-net in base 4, because

10 times m-reduction [i] would yield (138, 140, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (150−12, 150, s)-Nets in Base 4

(150−12, 150, 5592448)-Net over F4 — Constructive and digital

(150−12, 150, large)-Net over F4 — Digital

(150−12, 150, large)-Net in Base 4 — Upper bound on s

(150−12, 150, 5592448)-Net over F₄ — Constructive and digital

(150−12, 150, large)-Net over F₄ — Digital