MinT - Best Known (113, 113+30, s)-Nets in Base 4

Best Known (113, 113+30, s)-Nets in Base 4

(113, 113+30, 1049)-Net over F₄ — Constructive and digital

Digital (113, 143, 1049)-net over F₄, using

4¹ times duplication [i] based on digital (112, 142, 1049)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 22, 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 (90, 120, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 30, 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]

(113, 113+30, 4181)-Net over F₄ — Digital

Digital (113, 143, 4181)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴³, 4181, F₄, 30) (dual of [4181, 4038, 31]-code), using
- 69 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 16 times 0, 1, 26 times 0) [i] based on linear OA(4¹³³, 4102, F₄, 30) (dual of [4102, 3969, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
    1. linear OA(4¹³³, 4096, F₄, 30) (dual of [4096, 3963, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    2. linear OA(4¹²⁷, 4096, F₄, 29) (dual of [4096, 3969, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [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]

(113, 113+30, 1175654)-Net in Base 4 — Upper bound on s

There is no (113, 143, 1175655)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 124 332346 659523 472010 705293 518675 130726 647545 243231 891355 266057 688700 629626 258585 763504 > 4¹⁴³ [i]