MinT - Best Known (133−33, 133, s)-Nets in Base 5

Best Known (133−33, 133, s)-Nets in Base 5

(133−33, 133, 410)-Net over F₅ — Constructive and digital

Digital (100, 133, 410)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 10)-net over F₅, using
  - net from sequence [i] based on digital (1, 9)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 1 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (83, 116, 400)-net over F₅, using
  - trace code for nets [i] based on digital (25, 58, 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]

(133−33, 133, 2917)-Net over F₅ — Digital

Digital (100, 133, 2917)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹³³, 2917, F₅, 33) (dual of [2917, 2784, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹³³, 3138, F₅, 33) (dual of [3138, 3005, 34]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(5¹³², 3137, F₅, 33) (dual of [3137, 3005, 34]-code), using
    - constructionÂ X applied to C([0,16]) âŠ‚ C([0,15]) [i] based on
      1. linear OA(5¹³¹, 3126, F₅, 33) (dual of [3126, 2995, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      2. linear OA(5¹²¹, 3126, F₅, 31) (dual of [3126, 3005, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126Â | 5¹⁰âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
      3. linear OA(5¹, 11, F₅, 1) (dual of [11, 10, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(133−33, 133, 993061)-Net in Base 5 — Upper bound on s

There is no (100, 133, 993062)-net in base 5, because

1 times m-reduction [i] would yield (100, 132, 993062)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 183 671830 664931 236497 866206 389873 192180 705594 354744 355156 397075 778286 930957 678349 237067 007745 > 5¹³² [i]