MinT - Best Known (31−7, 31, s)-Nets in Base 64

Best Known (31−7, 31, s)-Nets in Base 64

(31−7, 31, 3066602)-Net over F₆₄ — Constructive and digital

Digital (24, 31, 3066602)-net over F₆₄, using

net defined by OOA [i] based on linear OOA(64³¹, 3066602, F₆₄, 9, 7) (dual of [(3066602, 9), 27599387, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(64³¹, 3066603, F₆₄, 3, 7) (dual of [(3066603, 3), 9199778, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(64⁶, 270402, F₆₄, 3, 3) (dual of [(270402, 3), 811200, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(64⁶, 270402, F₆₄, 2, 3) (dual of [(270402, 2), 540798, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁶, 270402, F₆₄, 3) (dual of [270402, 270396, 4]-code or 270402-cap in PG(5,64)), using
        large caps in PG(5,Â b) [i]
    2. linear OOA(64²⁵, 2796201, F₆₄, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(64²⁵, large, F₆₄, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(31−7, 31, large)-Net over F₆₄ — Digital

Digital (24, 31, large)-net over F₆₄, using

2 times m-reduction [i] based on digital (24, 33, large)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³³, large, F₆₄, 9) (dual of [large, large−33, 10]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(31−7, 31, large)-Net in Base 64 — Upper bound on s

There is no (24, 31, large)-net in base 64, because

5 times m-reduction [i] would yield (24, 26, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]