MinT - Best Known (61, 68, s)-Nets in Base 4

Best Known (61, 68, s)-Nets in Base 4

(61, 68, 2796488)-Net over F₄ — Constructive and digital

Digital (61, 68, 2796488)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁶⁸, 2796488, F₄, 9, 7) (dual of [(2796488, 9), 25168324, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(4⁶⁸, 2796489, F₄, 3, 7) (dual of [(2796489, 3), 8389399, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(4⁷, 288, F₄, 3, 3) (dual of [(288, 3), 857, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(4⁷, 288, F₄, 2, 3) (dual of [(288, 2), 569, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁷, 288, F₄, 3) (dual of [288, 281, 4]-code or 288-cap in PG(6,4)), using
        product of two caps with tangent hyperplane [i]
    2. linear OOA(4⁶¹, 2796201, F₄, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(4⁶¹, large, F₄, 7) (dual of [large, large−61, 8]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(61, 68, large)-Net over F₄ — Digital

Digital (61, 68, large)-net over F₄, using

4⁷ times duplication [i] based on digital (54, 61, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶¹, large, F₄, 7) (dual of [large, large−61, 8]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(61, 68, large)-Net in Base 4 — Upper bound on s

There is no (61, 68, large)-net in base 4, because

5 times m-reduction [i] would yield (61, 63, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]