Best Known (12, 17, s)-Nets in Base 4
(12, 17, 515)-Net over F4 — Constructive and digital
Digital (12, 17, 515)-net over F4, using
- net defined by OOA [i] based on linear OOA(417, 515, F4, 5, 5) (dual of [(515, 5), 2558, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [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
- OOA 2-folding and stacking with additional row [i] based on linear OA(417, 1031, F4, 5) (dual of [1031, 1014, 6]-code), using
(12, 17, 983)-Net over F4 — Digital
Digital (12, 17, 983)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(417, 983, F4, 5) (dual of [983, 966, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(417, 1030, F4, 5) (dual of [1030, 1013, 6]-code), using
- construction X 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(41, 6, F4, 1) (dual of [6, 5, 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 X applied to Ce(4) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(417, 1030, F4, 5) (dual of [1030, 1013, 6]-code), using
(12, 17, 30892)-Net in Base 4 — Upper bound on s
There is no (12, 17, 30893)-net in base 4, because
- 1 times m-reduction [i] would yield (12, 16, 30893)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 4295 022898 > 416 [i]