MinT - Best Known (88, 118, s)-Nets in Base 4

Best Known (88, 118, s)-Nets in Base 4

(88, 118, 531)-Net over F₄ — Constructive and digital

Digital (88, 118, 531)-net over F₄, using

t-expansion [i] based on digital (87, 118, 531)-net over F₄, using
- 2 times m-reduction [i] based on digital (87, 120, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 40, 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]

(88, 118, 576)-Net in Base 4 — Constructive

(88, 118, 576)-net in base 4, using

4¹ times duplication [i] based on (87, 117, 576)-net in base 4, using
- trace code for nets [i] based on (9, 39, 192)-net in base 64, using
  - 3 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
    - base change [i] based on digital (3, 36, 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]

(88, 118, 1123)-Net over F₄ — Digital

Digital (88, 118, 1123)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁸, 1123, F₄, 30) (dual of [1123, 1005, 31]-code), using
- 87 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 23 times 0, 1, 35 times 0) [i] based on linear OA(4¹¹¹, 1029, F₄, 30) (dual of [1029, 918, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
    1. linear OA(4¹¹¹, 1024, F₄, 30) (dual of [1024, 913, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    2. linear OA(4¹⁰⁶, 1024, F₄, 29) (dual of [1024, 918, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [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]

(88, 118, 116628)-Net in Base 4 — Upper bound on s

There is no (88, 118, 116629)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 110430 159096 985588 005411 184778 328824 505381 892359 984627 753718 787116 695184 > 4¹¹⁸ [i]