MinT - Best Known (121−12, 121, s)-Nets in Base 4

Best Known (121−12, 121, s)-Nets in Base 4

(121−12, 121, 1398135)-Net over F₄ — Constructive and digital

Digital (109, 121, 1398135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 13, 35)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 17)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4⁴, 17, F₄, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(4⁴, 17, F₄, 2, 3) (dual of [(17, 2), 30, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
        ovoid in PG(3,Â 4) [i]
    2. digital (3, 9, 18)-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]

(121−12, 121, large)-Net over F₄ — Digital

Digital (109, 121, large)-net over F₄, using

4⁷ times duplication [i] based on 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]

(121−12, 121, large)-Net in Base 4 — Upper bound on s

There is no (109, 121, large)-net in base 4, because

10 times m-reduction [i] would yield (109, 111, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]