MinT - Best Known (60, 60+33, s)-Nets in Base 16

Best Known (60, 60+33, s)-Nets in Base 16

(60, 60+33, 583)-Net over F₁₆ — Constructive and digital

Digital (60, 93, 583)-net over F₁₆, using

16¹ times duplication [i] based on digital (59, 92, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 22, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
  2. digital (37, 70, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 35, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(60, 60+33, 594)-Net in Base 16 — Constructive

(60, 93, 594)-net in base 16, using

16¹ times duplication [i] based on (59, 92, 594)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (10, 26, 80)-net in base 16, using
    - 1 times m-reduction [i] based on (10, 27, 80)-net in base 16, using
      - base change [i] based on digital (1, 18, 80)-net over F₆₄, using
        net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. digital (33, 66, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 33, 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]

(60, 60+33, 3084)-Net over F₁₆ — Digital

Digital (60, 93, 3084)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹³, 3084, F₁₆, 33) (dual of [3084, 2991, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹³, 4104, F₁₆, 33) (dual of [4104, 4011, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(29) [i] based on
    1. linear OA(16⁹¹, 4096, F₁₆, 33) (dual of [4096, 4005, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    2. linear OA(16⁸⁵, 4096, F₁₆, 30) (dual of [4096, 4011, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    3. linear OA(16², 8, F₁₆, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,16)), using
      - discarding factors / shortening the dual code based on linear OA(16², 16, F₁₆, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
        Reedâ€“Solomon code RS(14,16) [i]

(60, 60+33, 3803088)-Net in Base 16 — Upper bound on s

There is no (60, 93, 3803089)-net in base 16, because

1 times m-reduction [i] would yield (60, 92, 3803089)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 601 229146 011169 148538 690771 801068 227542 776583 736318 752418 627824 710934 540638 792091 007057 717832 969598 687485 563861 > 16⁹² [i]