MinT - Best Known (110−12, 110, s)-Nets in Base 16

Best Known (110−12, 110, s)-Nets in Base 16

(110−12, 110, 5625040)-Net over F₁₆ — Constructive and digital

Digital (98, 110, 5625040)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 11, 32640)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16¹¹, 32640, F₁₆, 4, 4) (dual of [(32640, 4), 130549, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(16¹¹, 65280, F₁₆, 4) (dual of [65280, 65269, 5]-code), using
      - base reduction for projective spaces (embedding PG(5,256) in PG(10,16)) [i] based on linear OA(256⁶, 65280, F₂₅₆, 4) (dual of [65280, 65274, 5]-code), using
        1 times truncation [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (25, 31, 2796200)-net over F₁₆, using
  - s-reduction based on digital (25, 31, 2796201)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16³¹, 2796201, F₁₆, 6, 6) (dual of [(2796201, 6), 16777175, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(16³¹, large, F₁₆, 6) (dual of [large, large−31, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. digital (56, 68, 2796200)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁶⁸, 2796200, F₁₆, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁶⁸, 8388601, F₁₆, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16⁶⁸, 8388602, F₁₆, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
        trace code [i] based on linear OOA(256³⁴, 4194301, F₂₅₆, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁴, 8388602, F₂₅₆, 12) (dual of [8388602, 8388568, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(110−12, 110, large)-Net over F₁₆ — Digital

Digital (98, 110, large)-net over F₁₆, using

t-expansion [i] based on digital (94, 110, large)-net over F₁₆, using
- 5 times m-reduction [i] based on digital (94, 115, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹¹⁵, large, F₁₆, 21) (dual of [large, large−115, 22]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(110−12, 110, large)-Net in Base 16 — Upper bound on s

There is no (98, 110, large)-net in base 16, because

10 times m-reduction [i] would yield (98, 100, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (110−12, 110, s)-Nets in Base 16

(110−12, 110, 5625040)-Net over F16 — Constructive and digital

(110−12, 110, large)-Net over F16 — Digital

(110−12, 110, large)-Net in Base 16 — Upper bound on s

(110−12, 110, 5625040)-Net over F₁₆ — Constructive and digital

(110−12, 110, large)-Net over F₁₆ — Digital