Best Known (208−33, 208, s)-Nets in Base 4
(208−33, 208, 4099)-Net over F4 — Constructive and digital
Digital (175, 208, 4099)-net over F4, using
- 42 times duplication [i] based on digital (173, 206, 4099)-net over F4, using
- net defined by OOA [i] based on linear OOA(4206, 4099, F4, 33, 33) (dual of [(4099, 33), 135061, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4206, 65585, F4, 33) (dual of [65585, 65379, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4206, 65589, F4, 33) (dual of [65589, 65383, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4153, 65536, F4, 26) (dual of [65536, 65383, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(413, 53, F4, 6) (dual of [53, 40, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4206, 65589, F4, 33) (dual of [65589, 65383, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4206, 65585, F4, 33) (dual of [65585, 65379, 34]-code), using
- net defined by OOA [i] based on linear OOA(4206, 4099, F4, 33, 33) (dual of [(4099, 33), 135061, 34]-NRT-code), using
(208−33, 208, 43338)-Net over F4 — Digital
Digital (175, 208, 43338)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4208, 43338, F4, 33) (dual of [43338, 43130, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4208, 65591, F4, 33) (dual of [65591, 65383, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4206, 65589, F4, 33) (dual of [65589, 65383, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4153, 65536, F4, 26) (dual of [65536, 65383, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(413, 53, F4, 6) (dual of [53, 40, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(32) ⊂ Ce(25) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4206, 65589, F4, 33) (dual of [65589, 65383, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4208, 65591, F4, 33) (dual of [65591, 65383, 34]-code), using
(208−33, 208, large)-Net in Base 4 — Upper bound on s
There is no (175, 208, large)-net in base 4, because
- 31 times m-reduction [i] would yield (175, 177, large)-net in base 4, but