MinT - Best Known (127−5, 127, s)-Nets in Base 16

Best Known (127−5, 127, s)-Nets in Base 16

(127−5, 127, large)-Net over F₁₆ — Constructive and digital

Digital (122, 127, large)-net over F₁₆, using

t-expansion [i] based on digital (116, 127, large)-net over F₁₆, using
- 2 times m-reduction [i] based on digital (116, 129, large)-net over F₁₆, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 17)-net over F₁₆, using
      - kÂ =Â 2 construction [i]
    2. digital (0, 3, 17)-net over F₁₆, using
      - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
        Niederreiter sequence [i]
    3. digital (15, 19, 2796200)-net over F₁₆, using
      - s-reduction based on digital (15, 19, 4194301)-net over F₁₆, using
        net defined by OOA [i] based on linear OOA(16¹⁹, 4194301, F₁₆, 4, 4) (dual of [(4194301, 4), 16777185, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(16¹⁹, 8388602, F₁₆, 4) (dual of [8388602, 8388583, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(16¹⁹, large, F₁₆, 4) (dual of [large, large−19, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
    4. 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]
    5. digital (61, 74, 2796200)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16⁷⁴, 2796200, F₁₆, 14, 13) (dual of [(2796200, 14), 39146726, 14]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁷⁴, 8388601, F₁₆, 2, 13) (dual of [(8388601, 2), 16777128, 14]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁷⁴, 8388602, F₁₆, 2, 13) (dual of [(8388602, 2), 16777130, 14]-NRT-code), using
        trace code [i] based on linear OOA(256³⁷, 4194301, F₂₅₆, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁷, 8388602, F₂₅₆, 13) (dual of [8388602, 8388565, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁷, large, F₂₅₆, 13) (dual of [large, large−37, 14]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(127−5, 127, large)-Net in Base 16 — Upper bound on s

There is no (122, 127, large)-net in base 16, because

3 times m-reduction [i] would yield (122, 124, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (127−5, 127, s)-Nets in Base 16

(127−5, 127, large)-Net over F16 — Constructive and digital

(127−5, 127, large)-Net in Base 16 — Upper bound on s

(127−5, 127, large)-Net over F₁₆ — Constructive and digital