Best Known (226−33, 226, s)-Nets in Base 4
(226−33, 226, 16386)-Net over F4 — Constructive and digital
Digital (193, 226, 16386)-net over F4, using
- 43 times duplication [i] based on digital (190, 223, 16386)-net over F4, using
- net defined by OOA [i] based on linear OOA(4223, 16386, F4, 33, 33) (dual of [(16386, 33), 540515, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4223, 262177, F4, 33) (dual of [262177, 261954, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4222, 262176, F4, 33) (dual of [262176, 261954, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- linear OA(4217, 262144, F4, 33) (dual of [262144, 261927, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4222, 262176, F4, 33) (dual of [262176, 261954, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4223, 262177, F4, 33) (dual of [262177, 261954, 34]-code), using
- net defined by OOA [i] based on linear OOA(4223, 16386, F4, 33, 33) (dual of [(16386, 33), 540515, 34]-NRT-code), using
(226−33, 226, 125593)-Net over F4 — Digital
Digital (193, 226, 125593)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4226, 125593, F4, 2, 33) (dual of [(125593, 2), 250960, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4226, 131090, F4, 2, 33) (dual of [(131090, 2), 261954, 34]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4222, 131088, F4, 2, 33) (dual of [(131088, 2), 261954, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4222, 262176, F4, 33) (dual of [262176, 261954, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- linear OA(4217, 262144, F4, 33) (dual of [262144, 261927, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(4222, 262176, F4, 33) (dual of [262176, 261954, 34]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4222, 131088, F4, 2, 33) (dual of [(131088, 2), 261954, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4226, 131090, F4, 2, 33) (dual of [(131090, 2), 261954, 34]-NRT-code), using
(226−33, 226, large)-Net in Base 4 — Upper bound on s
There is no (193, 226, large)-net in base 4, because
- 31 times m-reduction [i] would yield (193, 195, large)-net in base 4, but