MinT - Best Known (40, 40+6, s)-Nets in Base 5

Best Known (40, 40+6, s)-Nets in Base 5

(40, 40+6, 2796267)-Net over F₅ — Constructive and digital

Digital (40, 46, 2796267)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 5, 66)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁵, 66, F₅, 3, 3) (dual of [(66, 3), 193, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(5⁵, 66, F₅, 2, 3) (dual of [(66, 2), 127, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5⁵, 66, F₅, 3) (dual of [66, 61, 4]-code or 66-cap in PG(4,5)), using
        caps completed using a computer search [i]
2. digital (35, 41, 2796201)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁴¹, 2796201, F₅, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(5⁴¹, large, F₅, 6) (dual of [large, large−41, 7]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(40, 40+6, large)-Net over F₅ — Digital

Digital (40, 46, large)-net over F₅, using

5³ times duplication [i] based on digital (37, 43, large)-net over F₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁴³, large, F₅, 6) (dual of [large, large−43, 7]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(5⁴¹, large, F₅, 6) (dual of [large, large−41, 7]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(40, 40+6, large)-Net in Base 5 — Upper bound on s

There is no (40, 46, large)-net in base 5, because

4 times m-reduction [i] would yield (40, 42, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]