Best Known (169, 169+33, s)-Nets in Base 4
(169, 169+33, 4098)-Net over F4 — Constructive and digital
Digital (169, 202, 4098)-net over F4, using
- 41 times duplication [i] based on digital (168, 201, 4098)-net over F4, using
- net defined by OOA [i] based on linear OOA(4201, 4098, F4, 33, 33) (dual of [(4098, 33), 135033, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4201, 65569, F4, 33) (dual of [65569, 65368, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 65570, F4, 33) (dual of [65570, 65369, 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, 31, F4, 3) (dual of [31, 26, 4]-code or 31-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, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(32) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4201, 65570, F4, 33) (dual of [65570, 65369, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4201, 65569, F4, 33) (dual of [65569, 65368, 34]-code), using
- net defined by OOA [i] based on linear OOA(4201, 4098, F4, 33, 33) (dual of [(4098, 33), 135033, 34]-NRT-code), using
(169, 169+33, 33133)-Net over F4 — Digital
Digital (169, 202, 33133)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4202, 33133, F4, 33) (dual of [33133, 32931, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4202, 65573, F4, 33) (dual of [65573, 65371, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4161, 65537, F4, 27) (dual of [65537, 65376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(49, 36, F4, 5) (dual of [36, 27, 6]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4202, 65573, F4, 33) (dual of [65573, 65371, 34]-code), using
(169, 169+33, large)-Net in Base 4 — Upper bound on s
There is no (169, 202, large)-net in base 4, because
- 31 times m-reduction [i] would yield (169, 171, large)-net in base 4, but