MinT - Best Known (30, 36, s)-Nets in Base 16

Best Known (30, 36, s)-Nets in Base 16

(30, 36, 5592659)-Net over F₁₆ — Constructive and digital

Digital (30, 36, 5592659)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 4, 257)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁴, 257, F₁₆, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(16⁴, 257, F₁₆, 2, 3) (dual of [(257, 2), 510, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(16⁴, 257, F₁₆, 3) (dual of [257, 253, 4]-code or 257-cap in PG(3,16)), using
        ovoid in PG(3,Â 16) [i]
2. digital (26, 32, 5592402)-net over F₁₆, using
  - trace code for nets [i] based on digital (10, 16, 2796201)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256¹⁶, 2796201, F₂₅₆, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(256¹⁶, large, F₂₅₆, 6) (dual of [large, large−16, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(30, 36, large)-Net over F₁₆ — Digital

Digital (30, 36, large)-net over F₁₆, using

1 times m-reduction [i] based on digital (30, 37, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16³⁷, large, F₁₆, 7) (dual of [large, large−37, 8]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 16¹²âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(30, 36, large)-Net in Base 16 — Upper bound on s

There is no (30, 36, large)-net in base 16, because

4 times m-reduction [i] would yield (30, 32, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]