Best Known (217−34, 217, s)-Nets in Base 4
(217−34, 217, 3858)-Net over F4 — Constructive and digital
Digital (183, 217, 3858)-net over F4, using
- 43 times duplication [i] based on digital (180, 214, 3858)-net over F4, using
- net defined by OOA [i] based on linear OOA(4214, 3858, F4, 34, 34) (dual of [(3858, 34), 130958, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4214, 65586, F4, 34) (dual of [65586, 65372, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4214, 65589, F4, 34) (dual of [65589, 65375, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(26) [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(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(33) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4214, 65589, F4, 34) (dual of [65589, 65375, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(4214, 65586, F4, 34) (dual of [65586, 65372, 35]-code), using
- net defined by OOA [i] based on linear OOA(4214, 3858, F4, 34, 34) (dual of [(3858, 34), 130958, 35]-NRT-code), using
(217−34, 217, 49367)-Net over F4 — Digital
Digital (183, 217, 49367)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4217, 49367, F4, 34) (dual of [49367, 49150, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4217, 65600, F4, 34) (dual of [65600, 65383, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(25) [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(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(416, 64, F4, 7) (dual of [64, 48, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- construction X applied to Ce(33) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4217, 65600, F4, 34) (dual of [65600, 65383, 35]-code), using
(217−34, 217, large)-Net in Base 4 — Upper bound on s
There is no (183, 217, large)-net in base 4, because
- 32 times m-reduction [i] would yield (183, 185, large)-net in base 4, but