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

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

(31−5, 31, 39092)-Net over F₅ — Constructive and digital

Digital (26, 31, 39092)-net over F₅, using

net defined by OOA [i] based on linear OOA(5³¹, 39092, F₅, 6, 5) (dual of [(39092, 6), 234521, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(5³¹, 39093, F₅, 2, 5) (dual of [(39093, 2), 78155, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(5³, 31, F₅, 2, 2) (dual of [(31, 2), 59, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(5³, 31, F₅, 2) (dual of [31, 28, 3]-code), using
        Hamming code H(3,5) [i]
    2. linear OOA(5²⁸, 39062, F₅, 2, 5) (dual of [(39062, 2), 78096, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(5²⁸, 78124, F₅, 5) (dual of [78124, 78096, 6]-code), using
        1 times truncation [i] based on linear OA(5²⁹, 78125, F₅, 6) (dual of [78125, 78096, 7]-code), using
        an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(31−5, 31, 144549)-Net over F₅ — Digital

Digital (26, 31, 144549)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(31−5, 31, large)-Net in Base 5 — Upper bound on s

There is no (26, 31, large)-net in base 5, because

3 times m-reduction [i] would yield (26, 28, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]