Best Known (23−4, 23, s)-Nets in Base 5
(23−4, 23, 39070)-Net over F5 — Constructive and digital
Digital (19, 23, 39070)-net over F5, using
- net defined by OOA [i] based on linear OOA(523, 39070, F5, 4, 4) (dual of [(39070, 4), 156257, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(523, 39070, F5, 3, 4) (dual of [(39070, 3), 117187, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(523, 78140, F5, 4) (dual of [78140, 78117, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(522, 78125, F5, 4) (dual of [78125, 78103, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(58, 78125, F5, 2) (dual of [78125, 78117, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(523, 78140, F5, 4) (dual of [78140, 78117, 5]-code), using
- appending kth column [i] based on linear OOA(523, 39070, F5, 3, 4) (dual of [(39070, 3), 117187, 5]-NRT-code), using
(23−4, 23, 103777)-Net over F5 — Digital
Digital (19, 23, 103777)-net over F5, using
- net defined by OOA [i] based on linear OOA(523, 103777, F5, 4, 4) (dual of [(103777, 4), 415085, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(523, 103777, F5, 3, 4) (dual of [(103777, 3), 311308, 5]-NRT-code), using
(23−4, 23, large)-Net in Base 5 — Upper bound on s
There is no (19, 23, large)-net in base 5, because
- 2 times m-reduction [i] would yield (19, 21, large)-net in base 5, but