Best Known (147−21, 147, s)-Nets in Base 4
(147−21, 147, 26223)-Net over F4 — Constructive and digital
Digital (126, 147, 26223)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (115, 136, 26214)-net over F4, using
- net defined by OOA [i] based on linear OOA(4136, 26214, F4, 21, 21) (dual of [(26214, 21), 550358, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4136, 262141, F4, 21) (dual of [262141, 262005, 22]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4136, 262141, F4, 21) (dual of [262141, 262005, 22]-code), using
- net defined by OOA [i] based on linear OOA(4136, 26214, F4, 21, 21) (dual of [(26214, 21), 550358, 22]-NRT-code), using
- digital (1, 11, 9)-net over F4, using
(147−21, 147, 131095)-Net over F4 — Digital
Digital (126, 147, 131095)-net over F4, using
- 41 times duplication [i] based on digital (125, 146, 131095)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4146, 131095, F4, 2, 21) (dual of [(131095, 2), 262044, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4146, 262190, F4, 21) (dual of [262190, 262044, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- 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(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(410, 46, F4, 5) (dual of [46, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(4146, 262190, F4, 21) (dual of [262190, 262044, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4146, 131095, F4, 2, 21) (dual of [(131095, 2), 262044, 22]-NRT-code), using
(147−21, 147, large)-Net in Base 4 — Upper bound on s
There is no (126, 147, large)-net in base 4, because
- 19 times m-reduction [i] would yield (126, 128, large)-net in base 4, but