MinT - Best Known (151−12, 151, s)-Nets in Base 4

Best Known (151−12, 151, s)-Nets in Base 4

(151−12, 151, 5592452)-Net over F₄ — Constructive and digital

Digital (139, 151, 5592452)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 15, 52)-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 (4, 10, 26)-net over F₄, using
      - linear code embedding found in a computer search [i]
2. digital (124, 136, 5592400)-net over F₄, using
  - trace code for nets [i] based on digital (56, 68, 2796200)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁶⁸, 2796200, F₁₆, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁶⁸, 8388601, F₁₆, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁶⁸, 8388602, F₁₆, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
        trace code [i] based on linear OOA(256³⁴, 4194301, F₂₅₆, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁴, 8388602, F₂₅₆, 12) (dual of [8388602, 8388568, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(151−12, 151, large)-Net over F₄ — Digital

Digital (139, 151, large)-net over F₄, using

4 times m-reduction [i] based on digital (139, 155, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁵⁵, large, F₄, 16) (dual of [large, large−155, 17]-code), using
  - 11 times code embedding in larger space [i] based on linear OA(4¹⁴⁴, large, F₄, 16) (dual of [large, large−144, 17]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(151−12, 151, large)-Net in Base 4 — Upper bound on s

There is no (139, 151, large)-net in base 4, because

10 times m-reduction [i] would yield (139, 141, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (151−12, 151, s)-Nets in Base 4

(151−12, 151, 5592452)-Net over F4 — Constructive and digital

(151−12, 151, large)-Net over F4 — Digital

(151−12, 151, large)-Net in Base 4 — Upper bound on s

(151−12, 151, 5592452)-Net over F₄ — Constructive and digital

(151−12, 151, large)-Net over F₄ — Digital