MinT - Best Known (27−11, 27, s)-Nets in Base 5

Best Known (27−11, 27, s)-Nets in Base 5

(27−11, 27, 60)-Net over F₅ — Constructive and digital

Digital (16, 27, 60)-net over F₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 4, 20)-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]
2. digital (2, 7, 20)-net over F₅, using
  - s-reduction based on digital (2, 7, 21)-net over F₅, using
    - linear code embedding found in a computer search [i]
3. digital (5, 16, 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]

(27−11, 27, 103)-Net over F₅ — Digital

Digital (16, 27, 103)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5²⁷, 103, F₅, 11) (dual of [103, 76, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(5²⁷, 124, F₅, 11) (dual of [124, 97, 12]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 124Â = 5³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(27−11, 27, 2804)-Net in Base 5 — Upper bound on s

There is no (16, 27, 2805)-net in base 5, because

1 times m-reduction [i] would yield (16, 26, 2805)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 490377 937739 016965 > 5²⁶ [i]