Best Known (174−22, 174, s)-Nets in Base 4
(174−22, 174, 95336)-Net over F4 — Constructive and digital
Digital (152, 174, 95336)-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 (139, 161, 95326)-net over F4, using
- net defined by OOA [i] based on linear OOA(4161, 95326, F4, 22, 22) (dual of [(95326, 22), 2097011, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4161, 1048586, F4, 22) (dual of [1048586, 1048425, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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
- OA 11-folding and stacking [i] based on linear OA(4161, 1048586, F4, 22) (dual of [1048586, 1048425, 23]-code), using
- net defined by OOA [i] based on linear OOA(4161, 95326, F4, 22, 22) (dual of [(95326, 22), 2097011, 23]-NRT-code), using
- digital (2, 13, 10)-net over F4, using
(174−22, 174, 524319)-Net over F4 — Digital
Digital (152, 174, 524319)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4174, 524319, F4, 2, 22) (dual of [(524319, 2), 1048464, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4174, 1048638, F4, 22) (dual of [1048638, 1048464, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4174, 1048639, F4, 22) (dual of [1048639, 1048465, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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]
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4174, 1048639, F4, 22) (dual of [1048639, 1048465, 23]-code), using
- OOA 2-folding [i] based on linear OA(4174, 1048638, F4, 22) (dual of [1048638, 1048464, 23]-code), using
(174−22, 174, large)-Net in Base 4 — Upper bound on s
There is no (152, 174, large)-net in base 4, because
- 20 times m-reduction [i] would yield (152, 154, large)-net in base 4, but