Best Known (181, 181+37, s)-Nets in Base 4
(181, 181+37, 3641)-Net over F4 — Constructive and digital
Digital (181, 218, 3641)-net over F4, using
- net defined by OOA [i] based on linear OOA(4218, 3641, F4, 37, 37) (dual of [(3641, 37), 134499, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4218, 65539, F4, 37) (dual of [65539, 65321, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4218, 65545, F4, 37) (dual of [65545, 65327, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 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(41, 9, F4, 1) (dual of [9, 8, 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(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4218, 65545, F4, 37) (dual of [65545, 65327, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4218, 65539, F4, 37) (dual of [65539, 65321, 38]-code), using
(181, 181+37, 30117)-Net over F4 — Digital
Digital (181, 218, 30117)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4218, 30117, F4, 2, 37) (dual of [(30117, 2), 60016, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4218, 32772, F4, 2, 37) (dual of [(32772, 2), 65326, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4218, 65544, F4, 37) (dual of [65544, 65326, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4218, 65545, F4, 37) (dual of [65545, 65327, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 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(41, 9, F4, 1) (dual of [9, 8, 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(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4218, 65545, F4, 37) (dual of [65545, 65327, 38]-code), using
- OOA 2-folding [i] based on linear OA(4218, 65544, F4, 37) (dual of [65544, 65326, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(4218, 32772, F4, 2, 37) (dual of [(32772, 2), 65326, 38]-NRT-code), using
(181, 181+37, large)-Net in Base 4 — Upper bound on s
There is no (181, 218, large)-net in base 4, because
- 35 times m-reduction [i] would yield (181, 183, large)-net in base 4, but