Best Known (187, 187+37, s)-Nets in Base 4
(187, 187+37, 3642)-Net over F4 — Constructive and digital
Digital (187, 224, 3642)-net over F4, using
- 43 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
(187, 187+37, 32784)-Net over F4 — Digital
Digital (187, 224, 32784)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4224, 32784, F4, 2, 37) (dual of [(32784, 2), 65344, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4224, 65568, F4, 37) (dual of [65568, 65344, 38]-code), using
- construction XX applied to Ce(36) ⊂ Ce(32) ⊂ Ce(30) [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(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(45, 30, F4, 3) (dual of [30, 25, 4]-code or 30-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
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(36) ⊂ Ce(32) ⊂ Ce(30) [i] based on
- OOA 2-folding [i] based on linear OA(4224, 65568, F4, 37) (dual of [65568, 65344, 38]-code), using
(187, 187+37, large)-Net in Base 4 — Upper bound on s
There is no (187, 224, large)-net in base 4, because
- 35 times m-reduction [i] would yield (187, 189, large)-net in base 4, but