MinT - Best Known (87, 109, s)-Nets in Base 4

Best Known (87, 109, s)-Nets in Base 4

(87, 109, 1056)-Net over F₄ — Constructive and digital

Digital (87, 109, 1056)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 21, 28)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 7, 16)-net over F₄, using
      - linear code embedding found in a computer search [i]
    2. digital (3, 14, 14)-net over F₄, using
      - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, 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]

(87, 109, 4328)-Net over F₄ — Digital

Digital (87, 109, 4328)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰⁹, 4328, F₄, 22) (dual of [4328, 4219, 23]-code), using
- 214 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, 1, 50 times 0, 1, 77 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]

(87, 109, 1512716)-Net in Base 4 — Upper bound on s

There is no (87, 109, 1512717)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 421250 099625 157688 802863 153225 732080 303942 036819 474819 725234 282460 > 4¹⁰⁹ [i]