Best Known (200−33, 200, s)-Nets in Base 4
(200−33, 200, 4097)-Net over F4 — Constructive and digital
Digital (167, 200, 4097)-net over F4, using
- 44 times duplication [i] based on digital (163, 196, 4097)-net over F4, using
- net defined by OOA [i] based on linear OOA(4196, 4097, F4, 33, 33) (dual of [(4097, 33), 135005, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4196, 65553, F4, 33) (dual of [65553, 65357, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4196, 65555, F4, 33) (dual of [65555, 65359, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- 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(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(43, 19, F4, 2) (dual of [19, 16, 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
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4196, 65555, F4, 33) (dual of [65555, 65359, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4196, 65553, F4, 33) (dual of [65553, 65357, 34]-code), using
- net defined by OOA [i] based on linear OOA(4196, 4097, F4, 33, 33) (dual of [(4097, 33), 135005, 34]-NRT-code), using
(200−33, 200, 32784)-Net over F4 — Digital
Digital (167, 200, 32784)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4200, 32784, F4, 2, 33) (dual of [(32784, 2), 65368, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4200, 65568, F4, 33) (dual of [65568, 65368, 34]-code), using
- construction XX applied to Ce(32) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- 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(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(32) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- OOA 2-folding [i] based on linear OA(4200, 65568, F4, 33) (dual of [65568, 65368, 34]-code), using
(200−33, 200, large)-Net in Base 4 — Upper bound on s
There is no (167, 200, large)-net in base 4, because
- 31 times m-reduction [i] would yield (167, 169, large)-net in base 4, but