MinT - Best Known (123−15, 123, s)-Nets in Base 16

Best Known (123−15, 123, s)-Nets in Base 16

(123−15, 123, 4793484)-Net over F₁₆ — Constructive and digital

Digital (108, 123, 4793484)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (30, 37, 2796200)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16³⁷, 2796200, F₁₆, 7, 7) (dual of [(2796200, 7), 19573363, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(16³⁷, 8388601, F₁₆, 7) (dual of [8388601, 8388564, 8]-code), using
      - discarding factors / shortening the dual code 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]
2. digital (71, 86, 2396742)-net over F₁₆, using
  - trace code for nets [i] based on digital (28, 43, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴³, 1198371, F₂₅₆, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(256⁴³, 8388598, F₂₅₆, 15) (dual of [8388598, 8388555, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴³, large, F₂₅₆, 15) (dual of [large, large−43, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(123−15, 123, large)-Net over F₁₆ — Digital

Digital (108, 123, large)-net over F₁₆, using

t-expansion [i] based on digital (104, 123, large)-net over F₁₆, using
- 4 times m-reduction [i] based on digital (104, 127, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²⁷, large, F₁₆, 23) (dual of [large, large−127, 24]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(123−15, 123, large)-Net in Base 16 — Upper bound on s

There is no (108, 123, large)-net in base 16, because

13 times m-reduction [i] would yield (108, 110, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (123−15, 123, s)-Nets in Base 16

(123−15, 123, 4793484)-Net over F16 — Constructive and digital

(123−15, 123, large)-Net over F16 — Digital

(123−15, 123, large)-Net in Base 16 — Upper bound on s

(123−15, 123, 4793484)-Net over F₁₆ — Constructive and digital

(123−15, 123, large)-Net over F₁₆ — Digital