MinT - Best Known (144−38, 144, s)-Nets in Base 4

Best Known (144−38, 144, s)-Nets in Base 4

(144−38, 144, 531)-Net over F₄ — Constructive and digital

Digital (106, 144, 531)-net over F₄, using

t-expansion [i] based on digital (105, 144, 531)-net over F₄, using
- 3 times m-reduction [i] based on digital (105, 147, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 49, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(144−38, 144, 576)-Net in Base 4 — Constructive

(106, 144, 576)-net in base 4, using

trace code for nets [i] based on (10, 48, 192)-net in base 64, using
- 1 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
  - base change [i] based on digital (3, 42, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(144−38, 144, 1097)-Net over F₄ — Digital

Digital (106, 144, 1097)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴⁴, 1097, F₄, 38) (dual of [1097, 953, 39]-code), using
- 65 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 9 times 0, 1, 21 times 0, 1, 32 times 0) [i] based on linear OA(4¹⁴¹, 1029, F₄, 38) (dual of [1029, 888, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(36) [i] based on
    1. linear OA(4¹⁴¹, 1024, F₄, 38) (dual of [1024, 883, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(4¹³⁶, 1024, F₄, 37) (dual of [1024, 888, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    3. linear OA(4⁰, 5, F₄, 0) (dual of [5, 5, 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]

(144−38, 144, 96606)-Net in Base 4 — Upper bound on s

There is no (106, 144, 96607)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 497 372807 964835 025462 919365 643514 400123 376850 625344 687768 769480 698391 649437 006199 260632 > 4¹⁴⁴ [i]