Best Known (83, 99, s)-Nets in Base 4
(83, 99, 8194)-Net over F4 — Constructive and digital
Digital (83, 99, 8194)-net over F4, using
- 1 times m-reduction [i] based on digital (83, 100, 8194)-net over F4, using
- net defined by OOA [i] based on linear OOA(4100, 8194, F4, 17, 17) (dual of [(8194, 17), 139198, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4100, 65553, F4, 17) (dual of [65553, 65453, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4100, 65555, F4, 17) (dual of [65555, 65455, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(43, 19, F4, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4100, 65555, F4, 17) (dual of [65555, 65455, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4100, 65553, F4, 17) (dual of [65553, 65453, 18]-code), using
- net defined by OOA [i] based on linear OOA(4100, 8194, F4, 17, 17) (dual of [(8194, 17), 139198, 18]-NRT-code), using
(83, 99, 33008)-Net over F4 — Digital
Digital (83, 99, 33008)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(499, 33008, F4, 16) (dual of [33008, 32909, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(499, 65555, F4, 16) (dual of [65555, 65456, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(498, 65554, F4, 16) (dual of [65554, 65456, 17]-code), using
- construction X4 applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(417, 18, F4, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,4)), using
- dual of repetition code with length 18 [i]
- linear OA(41, 18, F4, 1) (dual of [18, 17, 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 X4 applied to Ce(16) ⊂ Ce(13) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(498, 65554, F4, 16) (dual of [65554, 65456, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(499, 65555, F4, 16) (dual of [65555, 65456, 17]-code), using
(83, 99, large)-Net in Base 4 — Upper bound on s
There is no (83, 99, large)-net in base 4, because
- 14 times m-reduction [i] would yield (83, 85, large)-net in base 4, but