MinT - Best Known (146−4, 146, s)-Nets in Base 5

Best Known (146−4, 146, s)-Nets in Base 5

(146−4, 146, large)-Net over F₅ — Constructive and digital

Digital (142, 146, large)-net over F₅, using

t-expansion [i] based on digital (139, 146, large)-net over F₅, using
- 3 times m-reduction [i] based on digital (139, 149, large)-net over F₅, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 6)-net over F₅, using
      - kÂ =Â 2 construction [i]
    2. digital (12, 15, 3050326)-net over F₅, using
      - net defined by OOA [i] based on linear OOA(5¹⁵, 3050326, F₅, 3, 3) (dual of [(3050326, 3), 9150963, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(5¹⁵, 3050326, F₅, 2, 3) (dual of [(3050326, 2), 6100637, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5¹⁵, 3050326, F₅, 3) (dual of [3050326, 3050311, 4]-code or 3050326-cap in PG(14,5)), using
        product of projective cap with cap in AG(6,Â 5) [i] based on linear OA(5⁹, 4700, F₅, 3) (dual of [4700, 4691, 4]-code or 4700-cap in PG(8,5)), using
        caps completed using a computer search [i]
    3. digital (35, 40, 3355440)-net over F₅, using
      - s-reduction based on digital (35, 40, 4194301)-net over F₅, using
        net defined by OOA [i] based on linear OOA(5⁴⁰, 4194301, F₅, 5, 5) (dual of [(4194301, 5), 20971465, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(5⁴⁰, large, F₅, 5) (dual of [large, large−40, 6]-code), using
        the primitive narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
    4. digital (82, 92, 3355440)-net over F₅, using
      - trace code for nets [i] based on digital (36, 46, 1677720)-net over F₂₅, using
        net defined by OOA [i] based on linear OOA(25⁴⁶, 1677720, F₂₅, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
        OA 5-folding and stacking [i] based on linear OA(25⁴⁶, 8388600, F₂₅, 10) (dual of [8388600, 8388554, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁴⁶, large, F₂₅, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(146−4, 146, large)-Net in Base 5 — Upper bound on s

There is no (142, 146, large)-net in base 5, because

2 times m-reduction [i] would yield (142, 144, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (146−4, 146, s)-Nets in Base 5

(146−4, 146, large)-Net over F5 — Constructive and digital

(146−4, 146, large)-Net in Base 5 — Upper bound on s

(146−4, 146, large)-Net over F₅ — Constructive and digital