Best Known (126−22, 126, s)-Nets in Base 4
(126−22, 126, 1500)-Net over F4 — Constructive and digital
Digital (104, 126, 1500)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 13, 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 (91, 113, 1490)-net over F4, using
- net defined by OOA [i] based on linear OOA(4113, 1490, F4, 22, 22) (dual of [(1490, 22), 32667, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4113, 16390, F4, 22) (dual of [16390, 16277, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4113, 16391, F4, 22) (dual of [16391, 16278, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 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(4113, 16391, F4, 22) (dual of [16391, 16278, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4113, 16390, F4, 22) (dual of [16390, 16277, 23]-code), using
- net defined by OOA [i] based on linear OOA(4113, 1490, F4, 22, 22) (dual of [(1490, 22), 32667, 23]-NRT-code), using
- digital (2, 13, 10)-net over F4, using
(126−22, 126, 16019)-Net over F4 — Digital
Digital (104, 126, 16019)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4126, 16019, F4, 22) (dual of [16019, 15893, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4126, 16401, F4, 22) (dual of [16401, 16275, 23]-code), using
- (u, u+v)-construction [i] based on
- linear OA(413, 17, F4, 11) (dual of [17, 4, 12]-code), using
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4126, 16401, F4, 22) (dual of [16401, 16275, 23]-code), using
(126−22, 126, large)-Net in Base 4 — Upper bound on s
There is no (104, 126, large)-net in base 4, because
- 20 times m-reduction [i] would yield (104, 106, large)-net in base 4, but