MinT - Best Known (85, 85+22, s)-Nets in Base 4

Best Known (85, 85+22, s)-Nets in Base 4

(85, 85+22, 1049)-Net over F₄ — Constructive and digital

Digital (85, 107, 1049)-net over F₄, using

4¹ times duplication [i] based on digital (84, 106, 1049)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 18, 21)-net over F₄, using
    - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (66, 88, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 22, 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]

(85, 85+22, 4197)-Net over F₄ — Digital

Digital (85, 107, 4197)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰⁷, 4197, F₄, 22) (dual of [4197, 4090, 23]-code), using
- 85 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 20 times 0, 1, 32 times 0) [i] based on linear OA(4⁹⁷, 4102, F₄, 22) (dual of [4102, 4005, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
    1. 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]
    2. linear OA(4⁹¹, 4096, F₄, 21) (dual of [4096, 4005, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [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]

(85, 85+22, 1175686)-Net in Base 4 — Upper bound on s

There is no (85, 107, 1175687)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 26328 261098 494171 827396 768814 515692 141847 487670 858914 286551 601536 > 4¹⁰⁷ [i]