Best Known (169, 169+25, s)-Nets in Base 4
(169, 169+25, 87390)-Net over F4 — Constructive and digital
Digital (169, 194, 87390)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 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 (156, 181, 87381)-net over F4, using
- net defined by OOA [i] based on linear OOA(4181, 87381, F4, 25, 25) (dual of [(87381, 25), 2184344, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4181, 1048573, F4, 25) (dual of [1048573, 1048392, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4181, 1048573, F4, 25) (dual of [1048573, 1048392, 26]-code), using
- net defined by OOA [i] based on linear OOA(4181, 87381, F4, 25, 25) (dual of [(87381, 25), 2184344, 26]-NRT-code), using
- digital (1, 13, 9)-net over F4, using
(169, 169+25, 524319)-Net over F4 — Digital
Digital (169, 194, 524319)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4194, 524319, F4, 2, 25) (dual of [(524319, 2), 1048444, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4194, 1048638, F4, 25) (dual of [1048638, 1048444, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4194, 1048639, F4, 25) (dual of [1048639, 1048445, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(24) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4194, 1048639, F4, 25) (dual of [1048639, 1048445, 26]-code), using
- OOA 2-folding [i] based on linear OA(4194, 1048638, F4, 25) (dual of [1048638, 1048444, 26]-code), using
(169, 169+25, large)-Net in Base 4 — Upper bound on s
There is no (169, 194, large)-net in base 4, because
- 23 times m-reduction [i] would yield (169, 171, large)-net in base 4, but