MinT - Best Known (40, 40+18, s)-Nets in Base 4

Best Known (40, 40+18, s)-Nets in Base 4

(40, 40+18, 240)-Net over F₄ — Constructive and digital

Digital (40, 58, 240)-net over F₄, using

4¹ times duplication [i] based on digital (39, 57, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 19, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(40, 40+18, 293)-Net over F₄ — Digital

Digital (40, 58, 293)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵⁸, 293, F₄, 18) (dual of [293, 235, 19]-code), using
- 28 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 11 times 0) [i] based on linear OA(4⁵¹, 258, F₄, 18) (dual of [258, 207, 19]-code), using
  - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
    1. linear OA(4⁵¹, 256, F₄, 18) (dual of [256, 205, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    2. linear OA(4⁴⁹, 256, F₄, 17) (dual of [256, 207, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(4⁰, 2, F₄, 0) (dual of [2, 2, 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]

(40, 40+18, 10478)-Net in Base 4 — Upper bound on s

There is no (40, 58, 10479)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 83141 193194 973785 883916 213274 579626 > 4⁵⁸ [i]