Best Known (159−22, 159, s)-Nets in Base 4
(159−22, 159, 23846)-Net over F4 — Constructive and digital
Digital (137, 159, 23846)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (123, 145, 23832)-net over F4, using
- net defined by OOA [i] based on linear OOA(4145, 23832, F4, 22, 22) (dual of [(23832, 22), 524159, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4145, 262152, F4, 22) (dual of [262152, 262007, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, 262153, F4, 22) (dual of [262153, 262008, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 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(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4145, 262153, F4, 22) (dual of [262153, 262008, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4145, 262152, F4, 22) (dual of [262152, 262007, 23]-code), using
- net defined by OOA [i] based on linear OOA(4145, 23832, F4, 22, 22) (dual of [(23832, 22), 524159, 23]-NRT-code), using
- digital (3, 14, 14)-net over F4, using
(159−22, 159, 157912)-Net over F4 — Digital
Digital (137, 159, 157912)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4159, 157912, F4, 22) (dual of [157912, 157753, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4159, 262203, F4, 22) (dual of [262203, 262044, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4158, 262202, F4, 22) (dual of [262202, 262044, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- 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(4100, 262144, F4, 15) (dual of [262144, 262044, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(21) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4158, 262202, F4, 22) (dual of [262202, 262044, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4159, 262203, F4, 22) (dual of [262203, 262044, 23]-code), using
(159−22, 159, large)-Net in Base 4 — Upper bound on s
There is no (137, 159, large)-net in base 4, because
- 20 times m-reduction [i] would yield (137, 139, large)-net in base 4, but