MinT - Best Known (90−18, 90, s)-Nets in Base 4

Best Known (90−18, 90, s)-Nets in Base 4

(90−18, 90, 1062)-Net over F₄ — Constructive and digital

Digital (72, 90, 1062)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 18, 34)-net over F₄, using
  - trace code for nets [i] based on digital (0, 9, 17)-net over F₁₆, using
    - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
      - Niederreiter sequence [i]
2. digital (54, 72, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 18, 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]

(90−18, 90, 4337)-Net over F₄ — Digital

Digital (72, 90, 4337)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹⁰, 4337, F₄, 18) (dual of [4337, 4247, 19]-code), using
- 224 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 12 times 0, 1, 21 times 0, 1, 34 times 0, 1, 54 times 0, 1, 82 times 0) [i] based on linear OA(4⁷⁹, 4102, F₄, 18) (dual of [4102, 4023, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
    1. linear OA(4⁷⁹, 4096, F₄, 18) (dual of [4096, 4017, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    2. linear OA(4⁷³, 4096, F₄, 17) (dual of [4096, 4023, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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]

(90−18, 90, 1449532)-Net in Base 4 — Upper bound on s

There is no (72, 90, 1449533)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 532498 638395 515280 926515 632639 963538 053557 317483 622752 > 4⁹⁰ [i]