MinT - Best Known (31−4, 31, s)-Nets in Base 5

Best Known (31−4, 31, s)-Nets in Base 5

Digital (27, 31, 4194301)-net over F₅, using

net defined by OOA [i] based on linear OOA(5³¹, 4194301, F₅, 4, 4) (dual of [(4194301, 4), 16777173, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(5³¹, 4194301, F₅, 3, 4) (dual of [(4194301, 3), 12582872, 5]-NRT-code), using
  - OA 2-folding and stacking [i] based on linear OA(5³¹, 8388602, F₅, 4) (dual of [8388602, 8388571, 5]-code), using
    - discarding factors / shortening the dual code based on linear OA(5³¹, large, F₅, 4) (dual of [large, large−31, 5]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

Digital (27, 31, large)-net over F₅, using

net defined by OOA [i] based on linear OOA(5³¹, large, F₅, 4, 4), using
- appending kth column [i] based on linear OOA(5³¹, large, F₅, 3, 4), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5³¹, large, F₅, 4) (dual of [large, large−31, 5]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

There is no (27, 31, large)-net in base 5, because