MinT - Best Known (112−23, 112, s)-Nets in Base 4

Best Known (112−23, 112, s)-Nets in Base 4

(112−23, 112, 1051)-Net over F₄ — Constructive and digital

Digital (89, 112, 1051)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 20, 23)-net over F₄, using
  - 3 times m-reduction [i] based on digital (9, 23, 23)-net over F₄, using
    - a shift-net [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]

(112−23, 112, 4206)-Net over F₄ — Digital

Digital (89, 112, 4206)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹², 4206, F₄, 23) (dual of [4206, 4094, 24]-code), using
- 95 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) [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]

(112−23, 112, 1946361)-Net in Base 4 — Upper bound on s

There is no (89, 112, 1946362)-net in base 4, because

1 times m-reduction [i] would yield (89, 111, 1946362)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 739988 302346 834354 013450 886932 897997 500300 939945 235877 536115 294536 > 4¹¹¹ [i]