MinT - Best Known (126−17, 126, s)-Nets in Base 16

Best Known (126−17, 126, s)-Nets in Base 16

(126−17, 126, 2098692)-Net over F₁₆ — Constructive and digital

Digital (109, 126, 2098692)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (20, 28, 1542)-net over F₁₆, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (2, 4, 514)-net over F₁₆, using
      - s-reduction based on digital (2, 4, 4369)-net over F₁₆, using
        kÂ =Â 2 construction [i]
    2. digital (4, 8, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 4, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
    3. digital (8, 16, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 8, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
2. digital (81, 98, 2097150)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁹⁸, 2097150, F₁₆, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(16⁹⁸, 8388601, F₁₆, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16⁹⁸, 8388602, F₁₆, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
        trace code [i] based on linear OOA(256⁴⁹, 4194301, F₂₅₆, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256⁴⁹, 8388602, F₂₅₆, 17) (dual of [8388602, 8388553, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁹, large, F₂₅₆, 17) (dual of [large, large−49, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(126−17, 126, 2105342)-Net in Base 16 — Constructive

(109, 126, 2105342)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (20, 28, 8192)-net in base 16, using
  - net defined by OOA [i] based on OOA(16²⁸, 8192, S₁₆, 8, 8), using
    - OA 4-folding and stacking [i] based on OA(16²⁸, 32768, S₁₆, 8), using
      - discarding factors based on OA(16²⁸, 32771, S₁₆, 8), using
        discarding parts of the base [i] based on linear OA(32²², 32771, F₃₂, 8) (dual of [32771, 32749, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(32²², 32768, F₃₂, 8) (dual of [32768, 32746, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(32¹⁹, 32768, F₃₂, 7) (dual of [32768, 32749, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(32⁰, 3, F₃₂, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (81, 98, 2097150)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁹⁸, 2097150, F₁₆, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(16⁹⁸, 8388601, F₁₆, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16⁹⁸, 8388602, F₁₆, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
        trace code [i] based on linear OOA(256⁴⁹, 4194301, F₂₅₆, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256⁴⁹, 8388602, F₂₅₆, 17) (dual of [8388602, 8388553, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁹, large, F₂₅₆, 17) (dual of [large, large−49, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(126−17, 126, large)-Net over F₁₆ — Digital

Digital (109, 126, large)-net over F₁₆, using

t-expansion [i] based on digital (104, 126, large)-net over F₁₆, using
- 1 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]

(126−17, 126, large)-Net in Base 16 — Upper bound on s

There is no (109, 126, large)-net in base 16, because

15 times m-reduction [i] would yield (109, 111, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (126−17, 126, s)-Nets in Base 16

(126−17, 126, 2098692)-Net over F16 — Constructive and digital

(126−17, 126, 2105342)-Net in Base 16 — Constructive

(126−17, 126, large)-Net over F16 — Digital

(126−17, 126, large)-Net in Base 16 — Upper bound on s

(126−17, 126, 2098692)-Net over F₁₆ — Constructive and digital

(126−17, 126, large)-Net over F₁₆ — Digital