Best Known (34, 34+7, s)-Nets in Base 4
(34, 34+7, 21847)-Net over F4 — Constructive and digital
Digital (34, 41, 21847)-net over F4, using
- net defined by OOA [i] based on linear OOA(441, 21847, F4, 7, 7) (dual of [(21847, 7), 152888, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(441, 65542, F4, 7) (dual of [65542, 65501, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(441, 65544, F4, 7) (dual of [65544, 65503, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(441, 65544, F4, 7) (dual of [65544, 65503, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(441, 65542, F4, 7) (dual of [65542, 65501, 8]-code), using
(34, 34+7, 56908)-Net over F4 — Digital
Digital (34, 41, 56908)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(441, 56908, F4, 7) (dual of [56908, 56867, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using
(34, 34+7, large)-Net in Base 4 — Upper bound on s
There is no (34, 41, large)-net in base 4, because
- 5 times m-reduction [i] would yield (34, 36, large)-net in base 4, but