Best Known (19, 19+5, s)-Nets in Base 4
(19, 19+5, 8196)-Net over F4 — Constructive and digital
Digital (19, 24, 8196)-net over F4, using
- 41 times duplication [i] based on digital (18, 23, 8196)-net over F4, using
- net defined by OOA [i] based on linear OOA(423, 8196, F4, 5, 5) (dual of [(8196, 5), 40957, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(423, 16393, F4, 5) (dual of [16393, 16370, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(415, 16384, F4, 3) (dual of [16384, 16369, 4]-code or 16384-cap in PG(14,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(48, 9, F4, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,4)), using
- dual of repetition code with length 9 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(423, 16393, F4, 5) (dual of [16393, 16370, 6]-code), using
- net defined by OOA [i] based on linear OOA(423, 8196, F4, 5, 5) (dual of [(8196, 5), 40957, 6]-NRT-code), using
(19, 19+5, 16395)-Net over F4 — Digital
Digital (19, 24, 16395)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(424, 16395, F4, 5) (dual of [16395, 16371, 6]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(423, 16393, F4, 5) (dual of [16393, 16370, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(415, 16384, F4, 3) (dual of [16384, 16369, 4]-code or 16384-cap in PG(14,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(48, 9, F4, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,4)), using
- dual of repetition code with length 9 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(423, 16394, F4, 4) (dual of [16394, 16371, 5]-code), using Gilbert–Varšamov bound and bm = 423 > Vbs−1(k−1) = 19 821423 675964 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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
- linear OA(423, 16393, F4, 5) (dual of [16393, 16370, 6]-code), using
- construction X with Varšamov bound [i] based on
(19, 19+5, 3954426)-Net in Base 4 — Upper bound on s
There is no (19, 24, 3954427)-net in base 4, because
- 1 times m-reduction [i] would yield (19, 23, 3954427)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 70 368759 563965 > 423 [i]