Best Known (82, 82+16, s)-Nets in Base 4
(82, 82+16, 8194)-Net over F4 — Constructive and digital
Digital (82, 98, 8194)-net over F4, using
- net defined by OOA [i] based on linear OOA(498, 8194, F4, 16, 16) (dual of [(8194, 16), 131006, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(498, 65552, F4, 16) (dual of [65552, 65454, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(498, 65553, F4, 16) (dual of [65553, 65455, 17]-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(41, 17, F4, 1) (dual of [17, 16, 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(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(498, 65553, F4, 16) (dual of [65553, 65455, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(498, 65552, F4, 16) (dual of [65552, 65454, 17]-code), using
(82, 82+16, 32777)-Net over F4 — Digital
Digital (82, 98, 32777)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(498, 32777, F4, 2, 16) (dual of [(32777, 2), 65456, 17]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(498, 65554, F4, 16) (dual of [65554, 65456, 17]-code), using
(82, 82+16, large)-Net in Base 4 — Upper bound on s
There is no (82, 98, large)-net in base 4, because
- 14 times m-reduction [i] would yield (82, 84, large)-net in base 4, but