Best Known (187−26, 187, s)-Nets in Base 4
(187−26, 187, 20175)-Net over F4 — Constructive and digital
Digital (161, 187, 20175)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 15, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (146, 172, 20165)-net over F4, using
- net defined by OOA [i] based on linear OOA(4172, 20165, F4, 26, 26) (dual of [(20165, 26), 524118, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4172, 262145, F4, 26) (dual of [262145, 261973, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4172, 262153, F4, 26) (dual of [262153, 261981, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(40, 9, F4, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4172, 262153, F4, 26) (dual of [262153, 261981, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4172, 262145, F4, 26) (dual of [262145, 261973, 27]-code), using
- net defined by OOA [i] based on linear OOA(4172, 20165, F4, 26, 26) (dual of [(20165, 26), 524118, 27]-NRT-code), using
- digital (2, 15, 10)-net over F4, using
(187−26, 187, 151409)-Net over F4 — Digital
Digital (161, 187, 151409)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4187, 151409, F4, 26) (dual of [151409, 151222, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4187, 262204, F4, 26) (dual of [262204, 262017, 27]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4185, 262202, F4, 26) (dual of [262202, 262017, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4127, 262144, F4, 19) (dual of [262144, 262017, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(413, 58, F4, 6) (dual of [58, 45, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4185, 262202, F4, 26) (dual of [262202, 262017, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4187, 262204, F4, 26) (dual of [262204, 262017, 27]-code), using
(187−26, 187, large)-Net in Base 4 — Upper bound on s
There is no (161, 187, large)-net in base 4, because
- 24 times m-reduction [i] would yield (161, 163, large)-net in base 4, but