Best Known (223−37, 223, s)-Nets in Base 4
(223−37, 223, 3642)-Net over F4 — Constructive and digital
Digital (186, 223, 3642)-net over F4, using
- 42 times duplication [i] based on digital (184, 221, 3642)-net over F4, using
- net defined by OOA [i] based on linear OOA(4221, 3642, F4, 37, 37) (dual of [(3642, 37), 134533, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(4221, 65557, F4, 37) (dual of [65557, 65336, 38]-code), using
- construction XX applied to Ce(36) ⊂ Ce(33) ⊂ Ce(32) [i] based on
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(43, 20, F4, 2) (dual of [20, 17, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(36) ⊂ Ce(33) ⊂ Ce(32) [i] based on
- OOA 18-folding and stacking with additional row [i] based on linear OA(4221, 65557, F4, 37) (dual of [65557, 65336, 38]-code), using
- net defined by OOA [i] based on linear OOA(4221, 3642, F4, 37, 37) (dual of [(3642, 37), 134533, 38]-NRT-code), using
(223−37, 223, 32783)-Net over F4 — Digital
Digital (186, 223, 32783)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4223, 32783, F4, 2, 37) (dual of [(32783, 2), 65343, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4223, 65566, F4, 37) (dual of [65566, 65343, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4222, 65565, F4, 37) (dual of [65565, 65343, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(32) [i] based on
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-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(36) ⊂ Ce(32) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4222, 65565, F4, 37) (dual of [65565, 65343, 38]-code), using
- OOA 2-folding [i] based on linear OA(4223, 65566, F4, 37) (dual of [65566, 65343, 38]-code), using
(223−37, 223, large)-Net in Base 4 — Upper bound on s
There is no (186, 223, large)-net in base 4, because
- 35 times m-reduction [i] would yield (186, 188, large)-net in base 4, but