MinT - Best Known (48−14, 48, s)-Nets in Base 4

Best Known (48−14, 48, s)-Nets in Base 4

(48−14, 48, 240)-Net over F₄ — Constructive and digital

Digital (34, 48, 240)-net over F₄, using

t-expansion [i] based on digital (33, 48, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 16, 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]

(48−14, 48, 328)-Net over F₄ — Digital

Digital (34, 48, 328)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴⁸, 328, F₄, 14) (dual of [328, 280, 15]-code), using
- 61 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 16 times 0, 1, 23 times 0) [i] based on linear OA(4⁴¹, 260, F₄, 14) (dual of [260, 219, 15]-code), using
  - constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
    1. linear OA(4⁴¹, 256, F₄, 14) (dual of [256, 215, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    2. linear OA(4³⁷, 256, F₄, 13) (dual of [256, 219, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [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]

(48−14, 48, 15137)-Net in Base 4 — Upper bound on s

There is no (34, 48, 15138)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 79244 379845 939024 912063 319760 > 4⁴⁸ [i]