Best Known (84, 84+16, s)-Nets in Base 4
(84, 84+16, 8194)-Net over F4 — Constructive and digital
Digital (84, 100, 8194)-net over F4, using
- t-expansion [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
(84, 84+16, 36445)-Net over F4 — Digital
Digital (84, 100, 36445)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4100, 36445, F4, 16) (dual of [36445, 36345, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4100, 65556, F4, 16) (dual of [65556, 65456, 17]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(497, 65537, F4, 17) (dual of [65537, 65440, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(481, 65537, F4, 13) (dual of [65537, 65456, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [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 C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4100, 65556, F4, 16) (dual of [65556, 65456, 17]-code), using
(84, 84+16, large)-Net in Base 4 — Upper bound on s
There is no (84, 100, large)-net in base 4, because
- 14 times m-reduction [i] would yield (84, 86, large)-net in base 4, but