MinT - Best Known (18−7, 18, s)-Nets in Base 5

Best Known (18−7, 18, s)-Nets in Base 5

(18−7, 18, 100)-Net over F₅ — Constructive and digital

Digital (11, 18, 100)-net over F₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 25)-net over F₅, using
  - s-reduction based on digital (0, 1, s)-net over F₅ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (1, 3, 25)-net over F₅, using
  - s-reduction based on digital (1, 3, 31)-net over F₅, using
    - kÂ =Â 2 construction [i]
3. digital (1, 4, 25)-net over F₅, using
  - s-reduction based on digital (1, 4, 26)-net over F₅, using
    - net defined by OOA [i] based on linear OOA(5⁴, 26, F₅, 3, 3) (dual of [(26, 3), 74, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(5⁴, 26, F₅, 2, 3) (dual of [(26, 2), 48, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5⁴, 26, F₅, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
        ovoid in PG(3,Â 5) [i]
4. digital (3, 10, 25)-net over F₅, using
  - linear code embedding found in a computer search [i]

(18−7, 18, 134)-Net over F₅ — Digital

Digital (11, 18, 134)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁸, 134, F₅, 7) (dual of [134, 116, 8]-code), using
- 4 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0) [i] based on linear OA(5¹⁶, 128, F₅, 7) (dual of [128, 112, 8]-code), using
  - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
    1. linear OA(5¹⁶, 125, F₅, 7) (dual of [125, 109, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 124Â = 5³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    2. linear OA(5¹³, 125, F₅, 6) (dual of [125, 112, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 124Â = 5³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
    3. linear OA(5⁰, 3, F₅, 0) (dual of [3, 3, 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]

(18−7, 18, 4149)-Net in Base 5 — Upper bound on s

There is no (11, 18, 4150)-net in base 5, because

1 times m-reduction [i] would yield (11, 17, 4150)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 763347 364601 > 5¹⁷ [i]