MinT - Best Known (15−4, 15, s)-Nets in Base 5

Best Known (15−4, 15, s)-Nets in Base 5

(15−4, 15, 781)-Net over F₅ — Constructive and digital

Digital (11, 15, 781)-net over F₅, using

net defined by OOA [i] based on linear OOA(5¹⁵, 781, F₅, 4, 4) (dual of [(781, 4), 3109, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(5¹⁵, 781, F₅, 3, 4) (dual of [(781, 3), 2328, 5]-NRT-code), using
  - OA 2-folding and stacking [i] based on linear OA(5¹⁵, 1562, F₅, 4) (dual of [1562, 1547, 5]-code), using
    - the BCH-code C(I) with length 1562Â | 5⁵âˆ’1, defining interval IÂ =Â {25,38,51,64}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(15−4, 15, 1660)-Net over F₅ — Digital

Digital (11, 15, 1660)-net over F₅, using

net defined by OOA [i] based on linear OOA(5¹⁵, 1660, F₅, 4, 4) (dual of [(1660, 4), 6625, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(5¹⁵, 1660, F₅, 3, 4) (dual of [(1660, 3), 4965, 5]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁵, 1660, F₅, 4) (dual of [1660, 1645, 5]-code), using
    - 457 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 19 times 0, 1, 93 times 0, 1, 342 times 0) [i] based on linear OA(5¹², 1200, F₅, 4) (dual of [1200, 1188, 5]-code), using
      - trace code [i] based on linear OA(25⁶, 600, F₂₅, 4) (dual of [600, 594, 5]-code), using
        1 times truncation [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]

(15−4, 15, 61762)-Net in Base 5 — Upper bound on s

There is no (11, 15, 61763)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 30518 333561 > 5¹⁵ [i]