MinT - Best Known (148−15, 148, s)-Nets in Base 4

Best Known (148−15, 148, s)-Nets in Base 4

(148−15, 148, 1198408)-Net over F₄ — Constructive and digital

Digital (133, 148, 1198408)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (8, 15, 37)-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 (4, 11, 20)-net over F₄, using
      - linear code embedding found in a computer search [i]
2. digital (118, 133, 1198371)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹³³, 1198371, F₄, 15, 15) (dual of [(1198371, 15), 17975432, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(4¹³³, 8388598, F₄, 15) (dual of [8388598, 8388465, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹³³, large, F₄, 15) (dual of [large, large−133, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(148−15, 148, large)-Net over F₄ — Digital

Digital (133, 148, large)-net over F₄, using

4³ times duplication [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]

(148−15, 148, large)-Net in Base 4 — Upper bound on s

There is no (133, 148, large)-net in base 4, because

13 times m-reduction [i] would yield (133, 135, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]