MinT - Best Known (57, 57+31, s)-Nets in Base 16

Best Known (57, 57+31, s)-Nets in Base 16

(57, 57+31, 583)-Net over F₁₆ — Constructive and digital

Digital (57, 88, 583)-net over F₁₆, using

16¹ times duplication [i] based on digital (56, 87, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 21, 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 (35, 66, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 33, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(57, 57+31, 596)-Net in Base 16 — Constructive

(57, 88, 596)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (9, 24, 80)-net in base 16, using
  - base change [i] based on digital (1, 16, 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, 64, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 32, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(57, 57+31, 3172)-Net over F₁₆ — Digital

Digital (57, 88, 3172)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁸, 3172, F₁₆, 31) (dual of [3172, 3084, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸⁸, 4096, F₁₆, 31) (dual of [4096, 4008, 32]-code), using
  - an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(57, 57+31, 4126391)-Net in Base 16 — Upper bound on s

There is no (57, 88, 4126392)-net in base 16, because

1 times m-reduction [i] would yield (57, 87, 4126392)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 573 375519 481267 599434 082825 951532 907628 718135 844574 373608 219519 087534 740772 686918 384307 883100 295383 607826 > 16⁸⁷ [i]