MinT - Best Known (82, 127, s)-Nets in Base 16

Best Known (82, 127, s)-Nets in Base 16

(82, 127, 587)-Net over F₁₆ — Constructive and digital

Digital (82, 127, 587)-net over F₁₆, using

16¹ times duplication [i] based on digital (81, 126, 587)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 28, 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 (53, 98, 522)-net over F₁₆, using
    - trace code for nets [i] based on digital (4, 49, 261)-net over F₂₅₆, using
      - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(82, 127, 612)-Net in Base 16 — Constructive

(82, 127, 612)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (15, 37, 98)-net in base 16, using
  - 3 times m-reduction [i] based on (15, 40, 98)-net in base 16, using
    - base change [i] based on digital (7, 32, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]
2. digital (45, 90, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 45, 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]

(82, 127, 3777)-Net over F₁₆ — Digital

Digital (82, 127, 3777)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²⁷, 3777, F₁₆, 45) (dual of [3777, 3650, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹²⁷, 4096, F₁₆, 45) (dual of [4096, 3969, 46]-code), using
  - an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]

(82, 127, 4754274)-Net in Base 16 — Upper bound on s

There is no (82, 127, 4754275)-net in base 16, because

1 times m-reduction [i] would yield (82, 126, 4754275)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 52 374480 823006 892813 981755 597344 846137 136620 909712 309016 584166 351392 923242 855963 056520 206604 618877 244532 872946 318542 744944 525717 971484 180762 410311 546376 > 16¹²⁶ [i]