Best Known (122, 122+23, s)-Nets in Base 4
(122, 122+23, 5961)-Net over F4 — Constructive and digital
Digital (122, 145, 5961)-net over F4, using
- 41 times duplication [i] based on digital (121, 144, 5961)-net over F4, using
- net defined by OOA [i] based on linear OOA(4144, 5961, F4, 23, 23) (dual of [(5961, 23), 136959, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4144, 65572, F4, 23) (dual of [65572, 65428, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4144, 65575, F4, 23) (dual of [65575, 65431, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(4137, 65536, F4, 23) (dual of [65536, 65399, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(47, 39, F4, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4144, 65575, F4, 23) (dual of [65575, 65431, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4144, 65572, F4, 23) (dual of [65572, 65428, 24]-code), using
- net defined by OOA [i] based on linear OOA(4144, 5961, F4, 23, 23) (dual of [(5961, 23), 136959, 24]-NRT-code), using
(122, 122+23, 38869)-Net over F4 — Digital
Digital (122, 145, 38869)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4145, 38869, F4, 23) (dual of [38869, 38724, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, 65576, F4, 23) (dual of [65576, 65431, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4144, 65575, F4, 23) (dual of [65575, 65431, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(4137, 65536, F4, 23) (dual of [65536, 65399, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(47, 39, F4, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4144, 65575, F4, 23) (dual of [65575, 65431, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, 65576, F4, 23) (dual of [65576, 65431, 24]-code), using
(122, 122+23, large)-Net in Base 4 — Upper bound on s
There is no (122, 145, large)-net in base 4, because
- 21 times m-reduction [i] would yield (122, 124, large)-net in base 4, but