MinT - Best Known (61, 61+6, s)-Nets in Base 5

Best Known (61, 61+6, s)-Nets in Base 5

(61, 61+6, large)-Net over F₅ — Constructive and digital

Digital (61, 67, large)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (46, 52, 5592402)-net over F₅, using
  - trace code for nets [i] based on digital (20, 26, 2796201)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25²⁶, 2796201, F₂₅, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(25²⁶, large, F₂₅, 6) (dual of [large, large−26, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(61, 61+6, large)-Net in Base 5 — Upper bound on s

There is no (61, 67, large)-net in base 5, because

4 times m-reduction [i] would yield (61, 63, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]