MinT - Best Known (61, 61+33, s)-Nets in Base 16

Best Known (61, 61+33, s)-Nets in Base 16

(61, 61+33, 585)-Net over F₁₆ — Constructive and digital

Digital (61, 94, 585)-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 (39, 72, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 36, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(61, 61+33, 643)-Net in Base 16 — Constructive

(61, 94, 643)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (12, 28, 129)-net in base 16, using
  - base change [i] based on digital (0, 16, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [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]

(61, 61+33, 3375)-Net over F₁₆ — Digital

Digital (61, 94, 3375)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁴, 3375, F₁₆, 33) (dual of [3375, 3281, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹⁴, 4106, F₁₆, 33) (dual of [4106, 4012, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,14]) [i] based on
    1. linear OA(16⁹¹, 4097, F₁₆, 33) (dual of [4097, 4006, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(16⁸⁵, 4097, F₁₆, 29) (dual of [4097, 4012, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 16⁶âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
    3. linear OA(16³, 9, F₁₆, 3) (dual of [9, 6, 4]-code or 9-arc in PG(2,16) or 9-cap in PG(2,16)), using
      - discarding factors / shortening the dual code based on linear OA(16³, 16, F₁₆, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
        Reedâ€“Solomon code RS(13,16) [i]

(61, 61+33, 4522661)-Net in Base 16 — Upper bound on s

There is no (61, 94, 4522662)-net in base 16, because

1 times m-reduction [i] would yield (61, 93, 4522662)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9619 662297 927226 394531 136065 346797 940769 436853 684409 139044 994107 466822 469096 427312 250034 184659 674522 124261 808006 > 16⁹³ [i]