MinT - Best Known (124−12, 124, s)-Nets in Base 4

Best Known (124−12, 124, s)-Nets in Base 4

(124−12, 124, 1398168)-Net over F₄ — Constructive and digital

Digital (112, 124, 1398168)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 16, 68)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 5, 41)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4⁵, 41, F₄, 3, 3) (dual of [(41, 3), 118, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(4⁵, 41, F₄, 2, 3) (dual of [(41, 2), 77, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]
    2. digital (5, 11, 34)-net over F₄, using
      - linear code embedding found in a computer search [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]

(124−12, 124, large)-Net over F₄ — Digital

Digital (112, 124, large)-net over F₄, using

t-expansion [i] based on digital (111, 124, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²⁴, large, F₄, 13) (dual of [large, large−124, 14]-code), using
  - 15 times code embedding in larger space [i] based on linear OA(4¹⁰⁹, large, F₄, 13) (dual of [large, large−109, 14]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(124−12, 124, large)-Net in Base 4 — Upper bound on s

There is no (112, 124, large)-net in base 4, because

10 times m-reduction [i] would yield (112, 114, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]