Best Known (219−35, 219, s)-Nets in Base 4
(219−35, 219, 3857)-Net over F4 — Constructive and digital
Digital (184, 219, 3857)-net over F4, using
- 43 times duplication [i] based on digital (181, 216, 3857)-net over F4, using
- net defined by OOA [i] based on linear OOA(4216, 3857, F4, 35, 35) (dual of [(3857, 35), 134779, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4216, 65570, F4, 35) (dual of [65570, 65354, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4216, 65575, F4, 35) (dual of [65575, 65359, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(29) [i] based on
- linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(47, 39, F4, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(34) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4216, 65575, F4, 35) (dual of [65575, 65359, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4216, 65570, F4, 35) (dual of [65570, 65354, 36]-code), using
- net defined by OOA [i] based on linear OOA(4216, 3857, F4, 35, 35) (dual of [(3857, 35), 134779, 36]-NRT-code), using
(219−35, 219, 41610)-Net over F4 — Digital
Digital (184, 219, 41610)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4219, 41610, F4, 35) (dual of [41610, 41391, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4219, 65579, F4, 35) (dual of [65579, 65360, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,14]) [i] based on
- linear OA(4209, 65537, F4, 35) (dual of [65537, 65328, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(4177, 65537, F4, 29) (dual of [65537, 65360, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(410, 42, F4, 5) (dual of [42, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to C([0,17]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4219, 65579, F4, 35) (dual of [65579, 65360, 36]-code), using
(219−35, 219, large)-Net in Base 4 — Upper bound on s
There is no (184, 219, large)-net in base 4, because
- 33 times m-reduction [i] would yield (184, 186, large)-net in base 4, but