Best Known (19, 19+6, s)-Nets in Base 4
(19, 19+6, 1367)-Net over F4 — Constructive and digital
Digital (19, 25, 1367)-net over F4, using
- net defined by OOA [i] based on linear OOA(425, 1367, F4, 6, 6) (dual of [(1367, 6), 8177, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(425, 4101, F4, 6) (dual of [4101, 4076, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(425, 4102, F4, 6) (dual of [4102, 4077, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(425, 4096, F4, 6) (dual of [4096, 4071, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(425, 4102, F4, 6) (dual of [4102, 4077, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(425, 4101, F4, 6) (dual of [4101, 4076, 7]-code), using
(19, 19+6, 3020)-Net over F4 — Digital
Digital (19, 25, 3020)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(425, 3020, F4, 6) (dual of [3020, 2995, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(425, 4096, F4, 6) (dual of [4096, 4071, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(425, 4096, F4, 6) (dual of [4096, 4071, 7]-code), using
(19, 19+6, 63010)-Net in Base 4 — Upper bound on s
There is no (19, 25, 63011)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1125 908201 826604 > 425 [i]