Best Known (156, 156+27, s)-Nets in Base 4
(156, 156+27, 20166)-Net over F4 — Constructive and digital
Digital (156, 183, 20166)-net over F4, using
- 41 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
(156, 156+27, 113424)-Net over F4 — Digital
Digital (156, 183, 113424)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4183, 113424, F4, 2, 27) (dual of [(113424, 2), 226665, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4183, 131082, F4, 2, 27) (dual of [(131082, 2), 261981, 28]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4182, 131082, F4, 2, 27) (dual of [(131082, 2), 261982, 28]-NRT-code), using
- OOA 2-folding [i] 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
- OOA 2-folding [i] based on linear OA(4182, 262164, F4, 27) (dual of [262164, 261982, 28]-code), using
- 41 times duplication [i] based on linear OOA(4182, 131082, F4, 2, 27) (dual of [(131082, 2), 261982, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4183, 131082, F4, 2, 27) (dual of [(131082, 2), 261981, 28]-NRT-code), using
(156, 156+27, large)-Net in Base 4 — Upper bound on s
There is no (156, 183, large)-net in base 4, because
- 25 times m-reduction [i] would yield (156, 158, large)-net in base 4, but