MinT - Best Known (161, 161+14, s)-Nets in Base 4

Best Known (161, 161+14, s)-Nets in Base 4

(161, 161+14, 4793521)-Net over F₄ — Constructive and digital

Digital (161, 175, 4793521)-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 (146, 160, 4793484)-net over F₄, using
  - trace code for nets [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(161, 161+14, large)-Net over F₄ — Digital

Digital (161, 175, large)-net over F₄, using

t-expansion [i] based on digital (158, 175, large)-net over F₄, using
- 1 times m-reduction [i] based on digital (158, 176, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷⁶, large, F₄, 18) (dual of [large, large−176, 19]-code), using
    - 19 times code embedding in larger space [i] based on linear OA(4¹⁵⁷, large, F₄, 18) (dual of [large, large−157, 19]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(161, 161+14, large)-Net in Base 4 — Upper bound on s

There is no (161, 175, large)-net in base 4, because

12 times m-reduction [i] would yield (161, 163, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]