MinT - Best Known (107, 144, s)-Nets in Base 4

Best Known (107, 144, s)-Nets in Base 4

(107, 144, 531)-Net over F₄ — Constructive and digital

Digital (107, 144, 531)-net over F₄, using

6 times m-reduction [i] based on digital (107, 150, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 50, 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]

(107, 144, 576)-Net in Base 4 — Constructive

(107, 144, 576)-net in base 4, using

t-expansion [i] based on (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]

(107, 144, 1237)-Net over F₄ — Digital

Digital (107, 144, 1237)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴⁴, 1237, F₄, 37) (dual of [1237, 1093, 38]-code), using
- 200 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0, 1, 10 times 0, 1, 23 times 0, 1, 33 times 0, 1, 39 times 0, 1, 42 times 0, 1, 44 times 0) [i] based on linear OA(4¹³⁷, 1030, F₄, 37) (dual of [1030, 893, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(34) [i] based on
    1. 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]
    2. linear OA(4¹³¹, 1024, F₄, 35) (dual of [1024, 893, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(4¹, 6, F₄, 1) (dual of [6, 5, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(107, 144, 152756)-Net in Base 4 — Upper bound on s

There is no (107, 144, 152757)-net in base 4, because

1 times m-reduction [i] would yield (107, 143, 152757)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 124 337951 479273 863600 250415 263201 421397 904530 484855 238880 291350 505949 320806 434802 310754 > 4¹⁴³ [i]