MinT - Best Known (34−10, 34, s)-Nets in Base 4

Best Known (34−10, 34, s)-Nets in Base 4

(34−10, 34, 240)-Net over F₄ — Constructive and digital

Digital (24, 34, 240)-net over F₄, using

4¹ times duplication [i] based on digital (23, 33, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 11, 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]

(34−10, 34, 298)-Net over F₄ — Digital

Digital (24, 34, 298)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³⁴, 298, F₄, 10) (dual of [298, 264, 11]-code), using
- 33 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 4 times 0, 1, 8 times 0, 1, 16 times 0) [i] based on linear OA(4²⁹, 260, F₄, 10) (dual of [260, 231, 11]-code), using
  - constructionÂ X applied to C_e(9) âŠ‚ C_e(8) [i] based on
    1. linear OA(4²⁹, 256, F₄, 10) (dual of [256, 227, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    2. linear OA(4²⁵, 256, F₄, 9) (dual of [256, 231, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(4⁰, 4, F₄, 0) (dual of [4, 4, 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]

(34−10, 34, 10778)-Net in Base 4 — Upper bound on s

There is no (24, 34, 10779)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 295 158860 125397 396884 > 4³⁴ [i]