Best Known (189, 189+30, s)-Nets in Base 4
(189, 189+30, 17493)-Net over F4 — Constructive and digital
Digital (189, 219, 17493)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 20, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (169, 199, 17476)-net over F4, using
- net defined by OOA [i] based on linear OOA(4199, 17476, F4, 30, 30) (dual of [(17476, 30), 524081, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4199, 262140, F4, 30) (dual of [262140, 261941, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4199, 262144, F4, 30) (dual of [262144, 261945, 31]-code), using
- an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(4199, 262144, F4, 30) (dual of [262144, 261945, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(4199, 262140, F4, 30) (dual of [262140, 261941, 31]-code), using
- net defined by OOA [i] based on linear OOA(4199, 17476, F4, 30, 30) (dual of [(17476, 30), 524081, 31]-NRT-code), using
- digital (5, 20, 17)-net over F4, using
(189, 189+30, 183357)-Net over F4 — Digital
Digital (189, 219, 183357)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4219, 183357, F4, 30) (dual of [183357, 183138, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4219, 262218, F4, 30) (dual of [262218, 261999, 31]-code), using
- 3 times code embedding in larger space [i] based on linear OA(4216, 262215, F4, 30) (dual of [262215, 261999, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(21) [i] based on
- linear OA(4199, 262144, F4, 30) (dual of [262144, 261945, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4145, 262144, F4, 22) (dual of [262144, 261999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(417, 71, F4, 7) (dual of [71, 54, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(416, 64, F4, 7) (dual of [64, 48, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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(6) ⊂ Ce(4) [i] based on
- construction X applied to Ce(29) ⊂ Ce(21) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(4216, 262215, F4, 30) (dual of [262215, 261999, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4219, 262218, F4, 30) (dual of [262218, 261999, 31]-code), using
(189, 189+30, large)-Net in Base 4 — Upper bound on s
There is no (189, 219, large)-net in base 4, because
- 28 times m-reduction [i] would yield (189, 191, large)-net in base 4, but