Best Known (18−5, 18, s)-Nets in Base 4
(18−5, 18, 516)-Net over F4 — Constructive and digital
Digital (13, 18, 516)-net over F4, using
- net defined by OOA [i] based on linear OOA(418, 516, F4, 6, 5) (dual of [(516, 6), 3078, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(418, 517, F4, 2, 5) (dual of [(517, 2), 1016, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(42, 5, F4, 2, 2) (dual of [(5, 2), 8, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(2;8,4) [i]
- linear OOA(416, 512, F4, 2, 5) (dual of [(512, 2), 1008, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(416, 1024, F4, 5) (dual of [1024, 1008, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- OOA 2-folding [i] based on linear OA(416, 1024, F4, 5) (dual of [1024, 1008, 6]-code), using
- linear OOA(42, 5, F4, 2, 2) (dual of [(5, 2), 8, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(418, 517, F4, 2, 5) (dual of [(517, 2), 1016, 6]-NRT-code), using
(18−5, 18, 992)-Net in Base 4 — Constructive
(13, 18, 992)-net in base 4, using
- net defined by OOA [i] based on OOA(418, 992, S4, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(418, 1985, S4, 5), using
- discarding factors based on OA(418, 1986, S4, 5), using
- trace code [i] based on OA(169, 993, S16, 5), using
- discarding parts of the base [i] based on linear OA(327, 993, F32, 5) (dual of [993, 986, 6]-code), using
- trace code [i] based on OA(169, 993, S16, 5), using
- discarding factors based on OA(418, 1986, S4, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(418, 1985, S4, 5), using
(18−5, 18, 1035)-Net over F4 — Digital
Digital (13, 18, 1035)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(418, 1035, F4, 5) (dual of [1035, 1017, 6]-code), using
- 3 step Varšamov–Edel lengthening with (ri) = (1, 0, 0) [i] based on linear OA(417, 1031, F4, 5) (dual of [1031, 1014, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(416, 1024, F4, 5) (dual of [1024, 1008, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(411, 1024, F4, 3) (dual of [1024, 1013, 4]-code or 1024-cap in PG(10,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(46, 7, F4, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,4)), using
- dual of repetition code with length 7 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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
- 3 step Varšamov–Edel lengthening with (ri) = (1, 0, 0) [i] based on linear OA(417, 1031, F4, 5) (dual of [1031, 1014, 6]-code), using
(18−5, 18, 61786)-Net in Base 4 — Upper bound on s
There is no (13, 18, 61787)-net in base 4, because
- 1 times m-reduction [i] would yield (13, 17, 61787)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 17179 998925 > 417 [i]