Best Known (179, 179+34, s)-Nets in Base 4
(179, 179+34, 3857)-Net over F4 — Constructive and digital
Digital (179, 213, 3857)-net over F4, using
- 45 times duplication [i] based on digital (174, 208, 3857)-net over F4, using
- net defined by OOA [i] based on linear OOA(4208, 3857, F4, 34, 34) (dual of [(3857, 34), 130930, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4208, 65569, F4, 34) (dual of [65569, 65361, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4208, 65575, F4, 34) (dual of [65575, 65367, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(33) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4208, 65575, F4, 34) (dual of [65575, 65367, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(4208, 65569, F4, 34) (dual of [65569, 65361, 35]-code), using
- net defined by OOA [i] based on linear OOA(4208, 3857, F4, 34, 34) (dual of [(3857, 34), 130930, 35]-NRT-code), using
(179, 179+34, 41509)-Net over F4 — Digital
Digital (179, 213, 41509)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4213, 41509, F4, 34) (dual of [41509, 41296, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4213, 65580, F4, 34) (dual of [65580, 65367, 35]-code), using
- 5 times code embedding in larger space [i] based on linear OA(4208, 65575, F4, 34) (dual of [65575, 65367, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(33) ⊂ Ce(28) [i] based on
- 5 times code embedding in larger space [i] based on linear OA(4208, 65575, F4, 34) (dual of [65575, 65367, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4213, 65580, F4, 34) (dual of [65580, 65367, 35]-code), using
(179, 179+34, large)-Net in Base 4 — Upper bound on s
There is no (179, 213, large)-net in base 4, because
- 32 times m-reduction [i] would yield (179, 181, large)-net in base 4, but