MinT - Best Known (111, 111+37, s)-Nets in Base 5

Best Known (111, 111+37, s)-Nets in Base 5

(111, 111+37, 460)-Net over F₅ — Constructive and digital

Digital (111, 148, 460)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (36, 54, 208)-net over F₅, using
  - trace code for nets [i] based on digital (9, 27, 104)-net over F₂₅, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
        a function field by GarcÃa and Quoos [i]
2. digital (57, 94, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 47, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]

(111, 111+37, 2974)-Net over F₅ — Digital

Digital (111, 148, 2974)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴⁸, 2974, F₅, 37) (dual of [2974, 2826, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹⁴⁸, 3133, F₅, 37) (dual of [3133, 2985, 38]-code), using
  - constructionÂ XX applied to C_e(36) âŠ‚ C_e(35) âŠ‚ C_e(33) [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(5¹³⁶, 3125, F₅, 34) (dual of [3125, 2989, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    4. linear OA(5⁰, 6, F₅, 0) (dual of [6, 6, 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]
    5. linear OA(5¹, 2, F₅, 1) (dual of [2, 1, 2]-code), using
      - dual of repetition code with length 2 [i]

(111, 111+37, 964537)-Net in Base 5 — Upper bound on s

There is no (111, 148, 964538)-net in base 5, because

1 times m-reduction [i] would yield (111, 147, 964538)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 5 605215 887759 861284 822120 924687 774952 342841 699739 404218 529486 397141 724484 313910 748121 803467 647712 050625 > 5¹⁴⁷ [i]

Best Known (111, 111+37, s)-Nets in Base 5

(111, 111+37, 460)-Net over F5 — Constructive and digital

(111, 111+37, 2974)-Net over F5 — Digital

(111, 111+37, 964537)-Net in Base 5 — Upper bound on s

(111, 111+37, 460)-Net over F₅ — Constructive and digital

(111, 111+37, 2974)-Net over F₅ — Digital