MinT - Best Known (114−13, 114, s)-Nets in Base 5

Best Known (114−13, 114, s)-Nets in Base 5

(114−13, 114, 1398153)-Net over F₅ — Constructive and digital

Digital (101, 114, 1398153)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (88, 101, 1398100)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹⁰¹, 1398100, F₅, 13, 13) (dual of [(1398100, 13), 18175199, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(5¹⁰¹, 8388601, F₅, 13) (dual of [8388601, 8388500, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹⁰¹, large, F₅, 13) (dual of [large, large−101, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 5²⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(114−13, 114, large)-Net over F₅ — Digital

Digital (101, 114, large)-net over F₅, using

5⁵ times duplication [i] based on digital (96, 109, large)-net over F₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁹, large, F₅, 13) (dual of [large, large−109, 14]-code), using
  - 8 times code embedding in larger space [i] based on linear OA(5¹⁰¹, large, F₅, 13) (dual of [large, large−101, 14]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 5²⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(114−13, 114, large)-Net in Base 5 — Upper bound on s

There is no (101, 114, large)-net in base 5, because

11 times m-reduction [i] would yield (101, 103, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]