Best Known (186, 186+33, s)-Nets in Base 4
(186, 186+33, 16384)-Net over F4 — Constructive and digital
Digital (186, 219, 16384)-net over F4, using
- 42 times duplication [i] based on digital (184, 217, 16384)-net over F4, using
- net defined by OOA [i] based on linear OOA(4217, 16384, F4, 33, 33) (dual of [(16384, 33), 540455, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4217, 262145, F4, 33) (dual of [262145, 261928, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(4217, 262145, F4, 33) (dual of [262145, 261928, 34]-code), using
- net defined by OOA [i] based on linear OOA(4217, 16384, F4, 33, 33) (dual of [(16384, 33), 540455, 34]-NRT-code), using
(186, 186+33, 90877)-Net over F4 — Digital
Digital (186, 219, 90877)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4219, 90877, F4, 2, 33) (dual of [(90877, 2), 181535, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4219, 131077, F4, 2, 33) (dual of [(131077, 2), 261935, 34]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4218, 131077, F4, 2, 33) (dual of [(131077, 2), 261936, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4218, 262154, F4, 33) (dual of [262154, 261936, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(4217, 262144, F4, 33) (dual of [262144, 261927, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(41, 10, F4, 1) (dual of [10, 9, 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 Ce(32) ⊂ Ce(30) [i] based on
- OOA 2-folding [i] based on linear OA(4218, 262154, F4, 33) (dual of [262154, 261936, 34]-code), using
- 41 times duplication [i] based on linear OOA(4218, 131077, F4, 2, 33) (dual of [(131077, 2), 261936, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4219, 131077, F4, 2, 33) (dual of [(131077, 2), 261935, 34]-NRT-code), using
(186, 186+33, large)-Net in Base 4 — Upper bound on s
There is no (186, 219, large)-net in base 4, because
- 31 times m-reduction [i] would yield (186, 188, large)-net in base 4, but