Best Known (185−27, 185, s)-Nets in Base 4
(185−27, 185, 20166)-Net over F4 — Constructive and digital
Digital (158, 185, 20166)-net over F4, using
- 43 times duplication [i] based on digital (155, 182, 20166)-net over F4, using
- net defined by OOA [i] based on linear OOA(4182, 20166, F4, 27, 27) (dual of [(20166, 27), 544300, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4182, 262159, F4, 27) (dual of [262159, 261977, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4182, 262164, F4, 27) (dual of [262164, 261982, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(4181, 262145, F4, 27) (dual of [262145, 261964, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(41, 19, F4, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4182, 262164, F4, 27) (dual of [262164, 261982, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4182, 262159, F4, 27) (dual of [262159, 261977, 28]-code), using
- net defined by OOA [i] based on linear OOA(4182, 20166, F4, 27, 27) (dual of [(20166, 27), 544300, 28]-NRT-code), using
(185−27, 185, 127316)-Net over F4 — Digital
Digital (158, 185, 127316)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4185, 127316, F4, 2, 27) (dual of [(127316, 2), 254447, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4185, 131084, F4, 2, 27) (dual of [(131084, 2), 261983, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4185, 262168, F4, 27) (dual of [262168, 261983, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4181, 262144, F4, 27) (dual of [262144, 261963, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4154, 262144, F4, 23) (dual of [262144, 261990, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, 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(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(4185, 262168, F4, 27) (dual of [262168, 261983, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(4185, 131084, F4, 2, 27) (dual of [(131084, 2), 261983, 28]-NRT-code), using
(185−27, 185, large)-Net in Base 4 — Upper bound on s
There is no (158, 185, large)-net in base 4, because
- 25 times m-reduction [i] would yield (158, 160, large)-net in base 4, but