MinT - Best Known (7, 7+6, s)-Nets in Base 5

Best Known (7, 7+6, s)-Nets in Base 5

(7, 7+6, 53)-Net over F₅ — Constructive and digital

Digital (7, 13, 53)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. 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 (3, 9, 27)-net over F₅, using
  - linear code embedding found in a computer search [i]

(7, 7+6, 67)-Net over F₅ — Digital

Digital (7, 13, 67)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹³, 67, F₅, 6) (dual of [67, 54, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹³, 124, F₅, 6) (dual of [124, 111, 7]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 124Â = 5³âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(7, 7+6, 483)-Net in Base 5 — Upper bound on s

There is no (7, 13, 484)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1222 531729 > 5¹³ [i]