MinT - Best Known (95, 95+23, s)-Nets in Base 4

Best Known (95, 95+23, s)-Nets in Base 4

(95, 95+23, 1094)-Net over F₄ — Constructive and digital

Digital (95, 118, 1094)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 26, 66)-net over F₄, using
  - trace code for nets [i] based on digital (2, 13, 33)-net over F₁₆, using
    - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
2. digital (69, 92, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 23, 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]

(95, 95+23, 5162)-Net over F₄ — Digital

Digital (95, 118, 5162)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁸, 5162, F₄, 23) (dual of [5162, 5044, 24]-code), using
- 1045 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 14 times 0, 1, 23 times 0, 1, 37 times 0, 1, 58 times 0, 1, 88 times 0, 1, 127 times 0, 1, 174 times 0, 1, 225 times 0, 1, 272 times 0) [i] based on linear OA(4¹⁰³, 4102, F₄, 23) (dual of [4102, 3999, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
    1. linear OA(4¹⁰³, 4096, F₄, 23) (dual of [4096, 3993, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    2. linear OA(4⁹⁷, 4096, F₄, 22) (dual of [4096, 3999, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(4⁰, 6, F₄, 0) (dual of [6, 6, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(95, 95+23, 4145920)-Net in Base 4 — Upper bound on s

There is no (95, 118, 4145921)-net in base 4, because

1 times m-reduction [i] would yield (95, 117, 4145921)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 27607 006452 386249 344344 240402 561359 723336 155705 541993 378544 502443 122944 > 4¹¹⁷ [i]