MinT - Best Known (147−37, 147, s)-Nets in Base 5

Best Known (147−37, 147, s)-Nets in Base 5

(147−37, 147, 420)-Net over F₅ — Constructive and digital

Digital (110, 147, 420)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 23, 20)-net over F₅, using
  - net from sequence [i] based on digital (5, 19)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 5 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (87, 124, 400)-net over F₅, using
  - trace code for nets [i] based on digital (25, 62, 200)-net over F₂₅, using
    - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
        a function field by GarcÃa and Quoos [i]

(147−37, 147, 2839)-Net over F₅ — Digital

Digital (110, 147, 2839)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴⁷, 2839, F₅, 37) (dual of [2839, 2692, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹⁴⁷, 3131, F₅, 37) (dual of [3131, 2984, 38]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(5¹⁴⁶, 3130, F₅, 37) (dual of [3130, 2984, 38]-code), using
    - constructionÂ X applied to C_e(36) âŠ‚ C_e(35) [i] based on
      1. linear OA(5¹⁴⁶, 3125, F₅, 37) (dual of [3125, 2979, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
      2. linear OA(5¹⁴¹, 3125, F₅, 36) (dual of [3125, 2984, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
      3. linear OA(5⁰, 5, F₅, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(5⁰, s, F₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(147−37, 147, 882037)-Net in Base 5 — Upper bound on s

There is no (110, 147, 882038)-net in base 5, because

1 times m-reduction [i] would yield (110, 146, 882038)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 121050 981169 403558 243552 617138 881399 051829 020578 004152 348176 432995 920609 649578 814636 140737 799928 650625 > 5¹⁴⁶ [i]