MinT - Best Known (31−13, 31, s)-Nets in Base 5

Best Known (31−13, 31, s)-Nets in Base 5

(31−13, 31, 56)-Net over F₅ — Constructive and digital

Digital (18, 31, 56)-net over F₅, using

1 times m-reduction [i] based on digital (18, 32, 56)-net over F₅, using
- trace code for nets [i] based on digital (2, 16, 28)-net over F₂₅, using
  - net from sequence [i] based on digital (2, 27)-sequence over F₂₅, using
    - Niederreiter sequence [i]

(31−13, 31, 93)-Net over F₅ — Digital

Digital (18, 31, 93)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5³¹, 93, F₅, 13) (dual of [93, 62, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(5³¹, 124, F₅, 13) (dual of [124, 93, 14]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 124Â = 5³âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(31−13, 31, 2334)-Net in Base 5 — Upper bound on s

There is no (18, 31, 2335)-net in base 5, because

1 times m-reduction [i] would yield (18, 30, 2335)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 931 549328 402238 856169 > 5³⁰ [i]