MinT - Best Known (109, 109+13, s)-Nets in Base 4

Best Known (109, 109+13, s)-Nets in Base 4

(109, 109+13, 1398135)-Net over F₄ — Constructive and digital

Digital (109, 122, 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, 109, 1398100)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁰⁹, 1398100, F₄, 13, 13) (dual of [(1398100, 13), 18175191, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(4¹⁰⁹, 8388601, F₄, 13) (dual of [8388601, 8388492, 14]-code), using
      - discarding factors / shortening the dual code 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]

(109, 109+13, 6863605)-Net over F₄ — Digital

Digital (109, 122, 6863605)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²², 6863605, F₄, 13) (dual of [6863605, 6863483, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹²², large, F₄, 13) (dual of [large, large−122, 14]-code), using
  - 13 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]

(109, 109+13, large)-Net in Base 4 — Upper bound on s

There is no (109, 122, large)-net in base 4, because

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