MinT - Best Known (104, 104+37, s)-Nets in Base 4

Best Known (104, 104+37, s)-Nets in Base 4

(104, 104+37, 531)-Net over F₄ — Constructive and digital

Digital (104, 141, 531)-net over F₄, using

t-expansion [i] based on digital (103, 141, 531)-net over F₄, using
- 3 times m-reduction [i] based on digital (103, 144, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 48, 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]

(104, 104+37, 576)-Net in Base 4 — Constructive

(104, 141, 576)-net in base 4, using

trace code for nets [i] based on (10, 47, 192)-net in base 64, using
- 2 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]

(104, 104+37, 1106)-Net over F₄ — Digital

Digital (104, 141, 1106)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴¹, 1106, F₄, 37) (dual of [1106, 965, 38]-code), using
- 72 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0, 1, 10 times 0, 1, 23 times 0, 1, 33 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]

(104, 104+37, 121239)-Net in Base 4 — Upper bound on s

There is no (104, 141, 121240)-net in base 4, because

1 times m-reduction [i] would yield (104, 140, 121240)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 942688 161116 599107 358441 156860 247983 627941 545317 695613 221750 202126 348527 534076 909465 > 4¹⁴⁰ [i]