MinT - Best Known (138, 138+8, s)-Nets in Base 4

Best Known (138, 138+8, s)-Nets in Base 4

(138, 138+8, large)-Net over F₄ — Constructive and digital

Digital (138, 146, large)-net over F₄, using

2 times m-reduction [i] based on digital (138, 148, large)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (32, 37, 4194301)-net over F₄, using
    - net defined by OOA [i] based on linear OOA(4³⁷, 4194301, F₄, 5, 5) (dual of [(4194301, 5), 20971468, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(4³⁷, large, F₄, 5) (dual of [large, large−37, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
  2. digital (101, 111, 5033160)-net over F₄, using
    - trace code for nets [i] based on digital (27, 37, 1677720)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64³⁷, 1677720, F₆₄, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
        OA 5-folding and stacking [i] based on linear OA(64³⁷, 8388600, F₆₄, 10) (dual of [8388600, 8388563, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(138, 138+8, large)-Net in Base 4 — Upper bound on s

There is no (138, 146, large)-net in base 4, because

6 times m-reduction [i] would yield (138, 140, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]